AXIOMATIC EQUILIBRIUM SELECTION
FOR GENERIC TWO-PLAYER GAMES
SRIHARI GOVINDAN AND ROBERT WILSON
Abstract. We impose three axioms on refinements of the Nash equilibria of games with
perfect recall that select connected closed nonempty subsets, called solutions.
Undominated Strategies: Each equilibrium in a solution uses undominated strategies.
Backward Induction: Each solution contains a quasi-perfect equilibrium.
Small Worlds: The solutions of a game are induced by the solutions of any larger game in
which it is embedded such that players’ strategies and payoffs are preserved.
For games with two players and generic payoffs these axioms characterize each solution
as an essential component of equilibria in undominated strategies, and thus a stable set as
defined by Mertens (1989).
1. Introduction
Nash’s [26, 27] definition of equilibrium is insufficient in some respects; e.g. an equilibrium
can use weakly dominated strategies, or yield an outcome different from the one predicted by
backward induction in a game with perfect information. The literature on refinements aims
to sharpen Nash’s definition to exclude such equilibria. Recent surveys include [10, 15, 36]
and we add comments in Section 6.
Kohlberg and Mertens [17] suggest that a refinement should be characterized by axioms
adapted from decision theory. They also specify properties that axioms should imply. Subsequently, Mertens [22, 23, 24, 25] defines the set-valued refinement called stability and shows
that it has these and other properties, including the following.1
1. Admissibility and Perfection. All equilibria in a stable set are perfect (hence admissible) so each player’s strategy in each equilibrium is undominated.
2. Backward Induction and Forward Induction. A stable set includes a proper equilibrium that induces a quasi-perfect (hence sequential) equilibrium in every extensiveform game with perfect recall that has the same normal form. A subset of a stable
Key words and phrases. game, equilibrium, refinement, axiom, admissibility, backward induction, small
worlds, stability. JEL subject classification: C72. Date: 4 October 2010.
1
In this paper our characterization in Theorem 5.2 suffices as the definition of a stable set. A simplified
rendition of Mertens’ [22] definition is that a connected closed set of equilibria is stable if it has the property
that the projection map, from a connected closed neighborhood in the graph of equilibria over the space of
players’ strategies perturbed toward mixed strategies, has a point of coincidence with every continuous map
having the same domain and range. Govindan and Mertens [4] establish an equivalent definition in terms of
players’ best-reply correspondences. Mertens’ general definition invokes homology theory and is used here
only in Appendices C (Lemma C.11) and D.
1
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SRIHARI GOVINDAN AND ROBERT WILSON
set survives iterative elimination of weakly dominated strategies and strategies that
are inferior replies at every equilibrium in the set.
3. Invariance, Small Worlds, and Decomposition. The stable sets of a game are the
projections of the stable sets of any larger game in which it is embedded. The stable
sets of the product of two independent games are the products of their stable sets.
For games in extensive form with perfect recall, we adopt three axioms that are versions of
these properties. We assume that a refinement selects for each game a nonempty collection
of nonempty connected closed subsets of equilibria, called solutions.2 Briefly, the axioms are
the following:
A. Undominated Strategies: A solution uses only undominated strategies.
B. Backward Induction: A solution contains a quasi-perfect equilibrium.
C. Small Worlds: A solution is immune to embedding the game within larger games
that preserve players’ strategies and payoffs.
In Axiom A we invoke only the implication of admissibility that no player uses a weakly
dominated strategy. These properties are equivalent in two-player games. Axiom B insists
on inclusion of a quasi-perfect equilibrium because it induces a sequential equilibrium for
which the continuation strategy from a player’s information set is admissible. Axiom C
ensures that a refinement is not vulnerable to framing effects depending on how the game
is presented within wider contexts called metagames. A metagame can include additional
players and additional pure strategies, provided these additional features do not alter optimal
decisions in the original game.
For games with two players and generic payoffs, we prove that these three axioms characterize refinements whose solutions are stable sets. Our characterization is cast in terms of the
‘enabling form’ of a game in which two pure strategies of a player are considered equivalent
if they exclude the same terminal nodes of the game tree — the enabling form is defined in
subsection 4.3 and explained further in Appendix A. Our main theorem establishes that the
axioms imply that each solution is an essential component of admissible equilibria.3 For the
enabling form of a game this is the defining property of a stable set.
When payoffs are generic, all equilibria in a component yield the same probability distribution over outcomes [19, Theorem 2]. For economic modeling and econometric studies,
therefore, the axioms’ chief implication is that a predicted outcome distribution should result
2
Solutions are assumed to be sets because Kohlberg and Mertens [17, pp. 1015, 1019, 1029] show that
there need not exist a single equilibrium satisfying weaker properties than the axioms invoked here. The
technical requirement that a solution is connected excludes the trivial refinement that selects all equilibria. If
only a single (possibly unconnected) subset is selected then only the trivial refinement satisfies the conditions
invoked by Norde, Potters, Reijnierse, and Vermeulen [28].
3
A component is a maximal closed connected set, and it is essential if it has the property described in
footnote 1. For the usual normal form of the game, a solution can be a subset of admissible equilibria in a
component that maps to a component of the enabling form of the game.
AXIOMATIC EQUILIBRIUM SELECTION
3
from an essential component of the game’s admissible equilibria. The secondary implication
that a solution includes all equilibria in the component is germane only for predicting players’
behaviors after deviations from equilibrium play, but it addresses the issue of whether sequential rationality is a relevant decision-theoretic criterion after deviations (Reny [31]). We
show that each equilibrium in a solution is induced by a quasi-perfect equilibrium in a solution of some metagame; therefore, it is sequentially rational when viewed in this metagame.
See [9, §2.3] for an example.
Section 2 establishes notation for Section 3, which specifies the axioms and a precise
definition of embedding a game in a larger game. Appendix B verifies that Nash equilibria
satisfy this definition of embedding. Section 4 establishes further notation and then Section
5 states and proves the main theorem, Theorem 5.1, using a convenient characterization of
stability stated in Theorem 5.2 and proved in Appendix C. Appendix D constructs a function
used in the proof, and Appendix E elaborates the assumption that payoffs are generic. The
proof is constructive in that each equilibrium in a stable set is shown to be induced by
a quasi-perfect equilibrium in a solution of a particular metagame with perfect recall that
embeds the given game. Section 6 interprets this result and provides concluding remarks.
2. Notation
This section provides sufficient notation for statements of the axioms in Section 3. Section
4 introduces additional notation for the theorems in Section 5 and Appendix C.
A typical game in extensive form is denoted Γ. Its specification includes a set 𝑁 of players,
a game tree with perfect recall for each player, and an assignment of players’ payoffs at each
terminal node of the tree. Let 𝐻𝑛 be player 𝑛’s collection of information sets, and let 𝐴𝑛 (ℎ)
be his set of feasible actions at information set ℎ ∈ 𝐻𝑛 . The specification of the tree can
include a completely mixed strategy of Nature.
A player’s pure strategy chooses an action at each of his information sets. Denote 𝑛’s
simplex of mixed strategies by Σ𝑛 and interpret its vertices as his set 𝑆𝑛 of pure strategies.
∏
∏
The sets of profiles of players’ pure and mixed strategies are 𝑆 = 𝑛 𝑆𝑛 and Σ = 𝑛 Σ𝑛 . The
normal form of Γ assigns to each profile of players’ pure strategies the profile of their expected
payoffs; equivalently, it is the multilinear (i.e. linear in each variable) function 𝐺 : Σ → ℝ𝑁
that to each profile of their mixed strategies assigns the profile of their expected payoffs.
A player’s behavioral strategy specifies a mixture over his actions at each of his information
∏
sets. Let 𝐵𝑛 be 𝑛’s set of behavioral strategies, and 𝐵 = 𝑛 𝐵𝑛 the set of profiles of players’
behavioral strategies. Each mixed strategy 𝜎𝑛 induces a behavioral strategy 𝑏𝑛 . Because the
game has perfect recall, for each behavioral profile there are profiles of mixed strategies that
induce it and yield the same distribution of outcomes (Kuhn [20]).
As defined by Nash [26, 27], an equilibrium is a profile of players’ mixed strategies such
that each player’s strategy is an optimal reply to other players’ strategies. That is, if
4
SRIHARI GOVINDAN AND ROBERT WILSON
BR𝑛 (𝜎) ≡ arg max𝜎𝑛′ ∈Σ𝑛 𝐺𝑛 (𝜎𝑛′ , 𝜎−𝑛 ) is player 𝑛’s best-reply correspondence, then 𝜎 ∈ Σ
is an equilibrium iff 𝜎𝑛 ∈ BR𝑛 (𝜎) for every player 𝑛. The analogous definition of equilibrium
in behavioral strategies is equivalent for games with perfect recall.
A refinement is a correspondence that assigns to each game a nonempty collection of
nonempty connected closed subsets of its equilibria. Each selected subset is called a solution.
According to the above definitions, a mixed or behavioral strategy makes choices even
at information sets that its previous choices exclude from being reached. In Section 4 we
consider pure strategies to be equivalent if they make the same choices at information sets
they do not exclude. And, we further simplify mixed and behavioral strategies by considering
only their induced probability distributions on non-excluded terminal nodes. The definitions
of an equilibrium and a refinement have equivalent statements in terms of these strategy
spaces. Each equilibrium in a reduced strategy space corresponds to a set of equilibria as
defined above, and analogously for solutions selected by a refinement. The axioms in Section
3 are stated in terms of mixed strategies. Because Axiom C implies invariance to redundant
strategies, later we use of equivalence classes of strategies.
3. The Axioms
3.1. Undominated Strategies. The first axiom requires simply that no player uses a
weakly dominated strategy. Say that a profile of players’ strategies is undominated if each
player’s strategy is undominated.
Axiom A (Undominated Strategies): Each equilibrium in a solution is undominated.
3.2. Backward Induction. We interpret sequential equilibrium as the generalization to
games with perfect recall of backward induction in games with perfect information, and
to be consistent with Axiom A we insist on conditionally admissible continuations from
information sets. Here we obtain these properties from quasi-perfect equilibrium.
The standard definition of quasi-perfect equilibrium relies on consideration of perturbed
strategies, but an alternative definition uses its representation as a lexicographic equilibrium
[1, 3]. In this decision-theoretic version, each player’s behavior is described initially by
a finite sequence of mixed strategies, interpreted as other players’ alternative hypotheses
about his actions, ordered lexicographically, but at a subsequent information set that reveals
deviation from equilibrium play, hypotheses that fail to explain the deviation are discarded.
Considering only a two-player game for simplicity, quasi-perfection requires that a player’s
continuation is lexicographically optimal against the remaining subsequence of the other
player. Thus, at each information set of a player he continues with the first strategy in his
sequence that reaches that information set, and this continuation must be lexicographically
optimal in reply to the subsequence consisting of those hypotheses about the other player
AXIOMATIC EQUILIBRIUM SELECTION
5
that enable the information set to be reached; viz., any alternative strategy that is superior in
reply to one such hypothesis is inferior in reply to some hypothesis earlier in the subsequence.
However, to preserve continuity with previous literature we rely here on the original definition by Van Damme [35, p. 8] and then derive the relevant lexicographic representation in
subsection 5.10 when proving Theorem 5.1.
Definition 3.1 (Quasi-Perfect Equilibrium). A profile 𝑏 ∈ 𝐵 of behavioral strategies is a
quasi-perfect equilibrium if it is the limit of a sequence of profiles of completely mixed behavioral strategies for which, for each player 𝑛, from each of his information sets, continuation
of his strategy 𝑏𝑛 is an optimal reply to every profile in the sequence.
Equivalently, if 𝐵𝑅𝑛 (⋅∣ℎ) is 𝑛’s best-reply correspondence in terms of behavioral strategies
that continue from his information set ℎ ∈ 𝐻𝑛 , and 𝑏𝑛 (ℎ) is the continuation of his behavioral
strategy 𝑏𝑛 from this information set, then the profile 𝑏 ∈ 𝐵 is quasi-perfect if
(∀ 𝑘) (∀ 𝑛 ∈ 𝑁, ℎ ∈ 𝐻𝑛 )
𝑏𝑛 (ℎ) ∈ 𝐵𝑅𝑛 (ˆ𝑏𝑘 ∣ℎ)
for some sequence ˆ𝑏𝑘 ∈ 𝐵 ∖ ∂𝐵 converging to 𝑏.
If the mixed-strategy profile 𝜎 ∈ Σ induces a behavioral profile 𝑏 ∈ 𝐵 that is a quasi-perfect
equilibrium then we say that 𝜎 too is quasi-perfect. Similarly, the justifying sequence ˆ𝑏𝑘 can
be represented by a sequence 𝜎
ˆ 𝑘 in Σ ∖ ∂Σ for which 𝜎
ˆ 𝑘 converges to a mixed strategy that
enables the same outcome distribution as 𝜎 does.
Van Damme shows that a quasi-perfect equilibrium induces a perfect equilibrium of the
normal form, and a sequential equilibrium of the extensive form. Moreover, by construction
a quasi-perfect equilibrium provides for each player an optimal continuation from each of his
information sets that is admissible [1]. If payoffs are generic then every sequential equilibrium
is extensive-form perfect [19, 2] and quasi-perfect [14, 29]. However, if payoffs are nongeneric
(as here in subsection 5.6) then quasi-perfection invokes a stronger form of sequential rationality than in Kreps and Wilson’s [19, §4,5] definition of sequential equilibrium. Sequential
equilibrium requires only that each player’s continuation from an information set is optimal given the belief that the other player continues according to the first strategy in the
subsequence enabling the information set to be reached, whereas quasi-perfection requires
lexicographic optimality against the entire subsequence.
The second axiom requires that some equilibrium in a solution is quasi-perfect.
Axiom B (Backward Induction): Each solution contains a quasi-perfect equilibrium.
If payoffs are generic then Axioms A and B imply that each solution lies in a component
of the undominated equilibria, each of which yields the same distribution over outcomes as
sequential equilibria in the solution.
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SRIHARI GOVINDAN AND ROBERT WILSON
3.3. Small Worlds. The third axiom requires that a refinement is not affected by extraneous features of wider contexts in which a game is embedded, provided such contexts do
not alter players’ feasible strategies and payoffs. An embedding allows the presence of additional players whose actions might provide the original players with additional pure strategies
equivalent to mixed strategies in the original game, and thus redundant. For simplicity, we
define an embedding using the normal form 𝐺 : Σ → ℝ𝑁 of the extensive-form game Γ.
˜ : Σ̃ × Σ𝑜 → ℝ𝑁 ∪ 𝑜 in which game 𝐺 is
An embedding is described by a ‘larger’ game 𝐺
˜ has outsiders
‘embedded,’ subject to certain restrictions specified below. The larger game 𝐺
in a set 𝑜, in addition to insiders who are the players in 𝑁 , and there can be additional
moves by Nature. An insider 𝑛 can have additional pure strategies in Σ̃𝑛 that are not pure
strategies in 𝑆𝑛 but are equivalent to mixed strategies in Σ𝑛 . The basic requirement is that
an embedding should preserve the game among insiders, conditional on actions by outsiders.
These restrictions have a technical formulation. There should exist a multilinear map
˜ 𝑛 = 𝐺𝑛 ∘ 𝑓 for each insider 𝑛. Moreover,
𝑓 : Σ̃ × Σ𝑜 → Σ that is surjective and such that 𝐺
to exclude an embedding from introducing correlation among insiders’ strategies, 𝑓 should
factor into separate multilinear maps (𝑓𝑛 )𝑛∈𝑁 , where each component is a map 𝑓𝑛 : Σ̃𝑛 ×Σ𝑜 →
Σ𝑛 such that 𝑓𝑛 (⋅, 𝜎𝑜 ) maps Σ̃𝑛 surjectively onto Σ𝑛 for each profile 𝜎𝑜 ∈ Σ𝑜 of outsiders’
strategies.
A statement of the axiom that uses this technical language could contain unsuspected implications, so after stating the formal definition we provide in Proposition 3.3 an equivalent
formulation that is more detailed and more transparent, and that verifies the requisite properties. Also, Proposition 3.4 applies a precise test of whether the axiom is correctly stated,
namely, a refinement that satisfies the axiom should be immune to the same embeddings
that equilibria are.
˜ : Σ̃ × Σ𝑜 → ℝ𝑁 ∪ 𝑜 and maps 𝑓 = (𝑓𝑛 )𝑛∈𝑁 , where
Definition 3.2 (Embedding). A game 𝐺
each map 𝑓𝑛 : Σ̃𝑛 × Σ𝑜 → Σ𝑛 is multilinear, embed a game 𝐺 : Σ → ℝ𝑁 if
(a) for each 𝜎𝑜 ∈ Σ𝑜 and 𝑛 ∈ 𝑁 , 𝑓𝑛 (⋅, 𝜎𝑜 ) maps Σ̃𝑛 surjectively onto Σ𝑛 , and
˜ = 𝐺 ∘ 𝑓.
(b) 𝐺
Condition (a) ensures that embedding has no net effect on an insider’s set of mixed strategies,
conditional on outsiders’ strategies, and condition (b) ensures that there is no net effect on
insiders’ payoffs.
˜ embeds 𝐺 via maps 𝑓 = (𝑓𝑛 ) then we say that (𝐺,
˜ 𝑓 ) embeds 𝐺 and that 𝐺
˜
Hereafter, if 𝐺
is a metagame for 𝐺. We omit description of 𝑓 for metagames in extensive form that embed
a game in extensive or strategic form. An elaborate example of a metagame in extensive
form that embeds a game in extensive form is constructed in proving Theorem 5.1.
A multilinear map 𝑓𝑛 : Σ̃𝑛 × Σ𝑜 → Σ𝑛 is completely specified by its values at profiles of
pure strategies. Let 𝑓ˆ𝑛 be the restriction of 𝑓𝑛 to the set 𝑆˜𝑛 × 𝑆𝑜 of profiles of pure strategies
AXIOMATIC EQUILIBRIUM SELECTION
7
of player 𝑛 and the outsiders in 𝑜, and let 𝑓ˆ = (𝑓ˆ𝑛 )𝑛∈𝑁 . The following proposition, proved
in Appendix B, provides an alternative definition of embedding in terms of pure strategies.
˜ 𝑓 ) embeds 𝐺 if and only if for each player 𝑛 there exists 𝑇˜𝑛 ⊆ 𝑆˜𝑛 and
Proposition 3.3. (𝐺,
a bijection 𝜋𝑛 : 𝑇˜𝑛 → 𝑆𝑛 such that for each (˜
𝑠, 𝑠𝑜 ) ∈ 𝑆˜ × 𝑆𝑜 and 𝑡˜𝑛 ∈ 𝑇˜𝑛 :
(1) 𝑓ˆ𝑛 (𝑡˜𝑛 , 𝑠𝑜 ) = 𝜋𝑛 (𝑡˜𝑛 ), and
˜ 𝑛 (˜
(2) 𝐺
𝑠, 𝑠𝑜 ) = 𝐺𝑛 (𝑓ˆ(˜
𝑠, 𝑠𝑜 )).
Property (2) assures that players’ payoffs from pure strategies of 𝐺 are preserved by the
˜ Hence property (1) assures that each pure strategy 𝑠𝑛 ∈ 𝑆𝑛 is equivalent to
metagame 𝐺.
some pure strategy 𝑡˜𝑛 = 𝜋 −1 (𝑠𝑛 ) ∈ 𝑇˜𝑛 independently of the outsiders’ profile 𝑠𝑜 .
∏
Pure strategies in 𝑆˜𝑛 ∖ 𝑇˜𝑛 are redundant because payoffs from profiles in 𝑛 𝑇˜𝑛 exactly
∏
replicate payoffs from corresponding profiles in 𝑛 𝑆𝑛 for the embedded game 𝐺. In particular, if 𝑓ˆ𝑛 (˜
𝑠𝑛 , 𝑠𝑜 ) = 𝜎𝑛 ∈
/ 𝑆𝑛 then conditional on 𝑠𝑜 the pure strategy 𝑠˜𝑛 is equivalent for
insiders to the mixed strategy 𝜎𝑛 ∈ Σ𝑛 .
The next Proposition, proved in Appendix B, verifies that equilibria are not affected by
embedding in a metagame.
˜ 𝑓 ) embeds 𝐺 then the equilibria of 𝐺 are the 𝑓 -images of the
Proposition 3.4. If (𝐺,
˜
equilibria of 𝐺.
A corollary of Proposition 3.4 is that embedding does not introduce correlation among
insiders’ strategies.
Using Definition 3.2 of embedding, the small worlds axiom is the following.
˜ 𝑓 ) embeds 𝐺 then the 𝑓 -images of the solutions that a
Axiom C (Small Worlds): If (𝐺,
˜ are the solutions selected for 𝐺.
refinement selects for 𝐺
In view of Proposition 3.4, this axiom is an instance of the general principle that a refinement
should inherit invariance properties of equilibria. Two special cases of Axiom C are the
following.
Invariance: Suppose Σ𝑜 and 𝑜 are singletons and insiders’ payoffs and strategies in Σ̃ differ
˜ Then
from Σ only by treating some mixed strategies in Σ as additional pure strategies in 𝑆.
Axiom C implies that solutions depend only on a game’s reduced normal form obtained by
deleting such redundant pure strategies.
Mertens’ Small Worlds Axiom [24]: Suppose Σ̃ = Σ. Then Axiom C implies that a
solution does not depend on the presence of outsiders, i.e. solutions of the original game are
the projections of the solutions of metagames obtained by adding players that are dummies
with respect to the game 𝐺.
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SRIHARI GOVINDAN AND ROBERT WILSON
3.4. Summary of the Axioms. We study refinements that are independent of embeddings
in metagames that, for each profile of outsiders’ strategies, preserve the strategies and payoffs
of the game among insiders. And, we require that each of their solutions is a closed connected
set of undominated equilibria, including one that is quasi-perfect. In particular, a solution
of a metagame must contain a quasi-perfect equilibrium whose image is in the corresponding
solution of the embedded game.
4. Additional Notation and Properties
In the sequel we consider only a game Γ in extensive form with perfect recall, two players,
and generic payoffs. In this section we prepare for the statement and proof of the main
theorem in Section 5.
4.1. Payoffs. Let 𝑍 be the set of terminal nodes of the game tree. Players’ payoffs are given
by a point 𝑢 in 𝑈 = ℝ𝑁 ×𝑍 , where 𝑢𝑛 (𝑧) is the payoff to player 𝑛 ∈ 𝑁 at terminal node 𝑧 ∈ 𝑍.
We assume that payoffs are generic in that there exists a lower dimensional subset 𝑈∘ of 𝑈
such that our results are true for all games in 𝑈 ∖ 𝑈∘ . The set 𝑈∘ includes the nongeneric
set described in [5]. Therefore, each game outside 𝑈∘ has finitely many equilibrium outcome
distributions, and in particular all equilibria in a component yield the same distribution
over outcomes. However, the proofs in Section 5 and Appendix C require some genericity
properties that are not necessarily implied by the construction in [5]. To avoid disrupting
the main exposition, we defer to Appendix E the description of the exact set of genericity
properties required for the proofs, and an explanation of why the resulting set 𝑈∘ of excluded
payoffs has lower dimension.
4.2. Notation for the Extensive Form. The set of players is 𝑁 = {1, 2}, typically represented as a player 𝑛 and the other player 𝑚 ∕= 𝑛. Let 𝑋 be the set of nodes in the game
tree. Let 𝑋𝑛 be the set of nodes where player 𝑛 moves, partitioned into his information
sets ℎ ∈ 𝐻𝑛 . For a node 𝑥 ∈ 𝑋𝑛 we write ℎ(𝑥) for the unique information set ℎ ∈ 𝐻𝑛 that
contains 𝑥. For each 𝑛 and ℎ ∈ 𝐻𝑛 , let 𝐴𝑛 (ℎ) be the set of actions available to player 𝑛 at
ℎ. Assume that actions at all information sets are labeled differently, and let 𝐴𝑛 be the set
of all actions for player 𝑛.
Node 𝑥 precedes another node 𝑦, written 𝑥 ≺ 𝑦, if 𝑥 is on the unique path from the root
of the tree to 𝑦. For a node 𝑥 ∈ 𝑋𝑛 and 𝑎 ∈ 𝐴𝑛 (ℎ(𝑥)) write (𝑥, 𝑎) ≺ 𝑦 if 𝑥 ≺ 𝑦 and the path
from the root of the tree to 𝑦 requires player 𝑛 to choose 𝑎 at ℎ(𝑥). If (𝑥, 𝑎) ≺ 𝑦 and 𝑥 and
𝑦 belong to 𝑛’s information sets ℎ and ℎ′ , respectively, then every node in ℎ′ follows some
node in ℎ by the choice of 𝑎, so we write (ℎ, 𝑎) ≺ ℎ′ .
The set of pure strategies of player 𝑛 is the set 𝑆𝑛 of functions 𝑠𝑛 : 𝐻𝑛 → 𝐴𝑛 such that
𝑠𝑛 (ℎ) ∈ 𝐴𝑛 (ℎ) for all ℎ ∈ 𝐻𝑛 . For each 𝑛, 𝑠𝑛 ∈ 𝑆𝑛 and 𝑦 ∈ 𝑋, let 𝛽𝑛 (𝑦, 𝑠𝑛 ) be the probability
that 𝑠𝑛 does not exclude 𝑦, i.e. 𝛽𝑛 (𝑦, 𝑠𝑛 ) = 1 if for each (𝑥, 𝑎) ≺ 𝑦 with 𝑥 ∈ 𝑋𝑛 , 𝑠𝑛 (ℎ(𝑥)) = 𝑎,
AXIOMATIC EQUILIBRIUM SELECTION
9
and otherwise 𝛽𝑛 (𝑦, 𝑠𝑛 ) = 0. By perfect recall, if 𝑦 ∈ 𝑋𝑛 then 𝛽𝑛 (𝑦 ′ , 𝑠𝑛 ) = 𝛽𝑛 (𝑦, 𝑠𝑛 ) for all
𝑦 ′ ∈ ℎ(𝑦) and we write 𝛽𝑛 (ℎ(𝑦), 𝑠𝑛 ) for this probability. Likewise, for any node 𝑦 we write
𝛽0 (𝑦) for the probability that Nature does not exclude 𝑦. Then for a profile 𝑠 ∈ 𝑆 the
probability that node 𝑦 is reached is 𝛽(𝑦, 𝑠) ≡ 𝛽0 (𝑦)𝛽1 (𝑦, 𝑠1 )𝛽2 (𝑦, 𝑠2 ).
Recall that Σ𝑛 is the set of mixed strategies of player 𝑛. For each node 𝑦 the function
∑
𝛽𝑛 (𝑦, ⋅) extends to a function over Σ𝑛 via 𝛽𝑛 (𝑦, 𝜎𝑛 ) = 𝑠𝑛 ∈𝑆𝑛 𝛽𝑛 (𝑦, 𝑠𝑛 )𝜎𝑛 (𝑠𝑛 ) for 𝜎𝑛 ∈ Σ𝑛 .
Recall also that 𝐵𝑛 is the set of behavioral strategies of player 𝑛. For each 𝑏𝑛 ∈ 𝐵𝑛 , 𝛽𝑛 (𝑦, 𝑏𝑛 )
is the product of 𝑏𝑛 ’s probabilities of 𝑛’s actions on the path to 𝑦.
Similarly extend 𝛽 to profiles of mixed or behavioral strategies. Given a mixed-strategy
profile 𝜎 ∈ Σ, the probability that outcome 𝑧 results is 𝛽(𝑧, 𝜎) = 𝛽0 (𝑧)𝛽1 (𝑧, 𝜎1 )𝛽2 (𝑧, 𝜎2 ).
4.3. Enabling Strategies. For each player 𝑛 define 𝜌𝑛 : Σ𝑛 → [0, 1]𝑍 by the formula
𝜌𝑛 (𝜎𝑛 ) = (𝛽𝑛 (𝑧, 𝜎𝑛 ))𝑧∈𝑍 , and let 𝜌 = (𝜌𝑛 )𝑛∈𝑁 . Similarly, if 𝑏𝑛 ∈ 𝐵𝑛 is the behavioral strategy
∏
induced by 𝜎𝑛 then 𝜌𝑛 (𝜎𝑛 ) = (𝛽𝑛 (𝑧, 𝑏𝑛 ))𝑧∈𝑍 . Let 𝑃𝑛 be the image of 𝜌𝑛 , and 𝑃 = 𝑛 𝑃𝑛
the image of 𝜌. Then 𝑃𝑛 is a compact convex polyhedron, called the space of 𝑛’s enabling
strategies in [6]. Each vertex of 𝑃𝑛 corresponds to an equivalence class of 𝑛’s pure strategies
that exclude the same outcomes. The vertices of 𝑃𝑛 are 𝑛’s pure strategies in the ‘pure
reduced normal form’ defined by Mailath, Samuelson, and Swinkels [21]; see Appendix A for
illustrations.
If 𝜎 ∈ Σ and 𝑝 = 𝜌(𝜎) then the probability of outcome 𝑧 is 𝛾𝑧 (𝑝) = 𝛽0 (𝑧)𝑝1 (𝑧)𝑝2 (𝑧). Thus
the function 𝛾 : 𝑃 → Δ(𝑍) summarizes the extensive form. The analog of the game Γ’s
normal form 𝐺 : Σ → ℝ𝑁 is the enabling form 𝒢 : 𝑃 → ℝ𝑁 that assigns to each profile
∑
of enabling strategies the profile of players’ expected payoffs, where 𝒢𝑛 (𝑝) = 𝑧 𝛾𝑧 (𝑝)𝑢𝑛 (𝑧).
Note that 𝛾 and 𝒢 are multilinear functions. From players’ best-reply correspondences in
terms of enabling strategies one obtains the definition of equilibrium in enabling strategies.
To each equilibrium in enabling strategies there correspond families of outcome-equivalent
equilibria in behavioral and mixed strategies. The axioms have direct analogs in terms of
enabling strategies, as shown in [12].
For games in extensive form with perfect recall, enabling strategies are minimal representations. For example, using perfect recall, by working backward in the induced tree of
a player’s information sets, from his enabling strategy one can construct the corresponding behavioral strategy at his information sets that his prior actions do not exclude [6, §5].
Because Axiom C implies Invariance, it is immaterial whether solutions are characterized
in terms of mixed or enabling strategies. We use enabling strategies here because induced
distributions over outcomes are multilinear functions of enabling strategies, like they are for
mixed strategies but unlike the nonlinear dependence on behavioral strategies. Also, the
10
SRIHARI GOVINDAN AND ROBERT WILSON
dimensions of the spaces of enabling and behavioral strategies are the same, which is important for the technical property established in Theorem 5.2 below. Using these features,
Section 5 derives the implications of the axioms in terms of enabling strategies.
5. Statement and Proof of the Theorem
We now state and prove the main theorem. As in Section 4 we consider a game Γ in
extensive form with perfect recall, two players, and generic payoffs. We assume that a
solution is represented in terms of enabling strategies, i.e. 𝑄∗ ⊂ 𝑃 is a solution iff it is the
image under 𝜌 of a solution Σ∗ ⊂ Σ. We say that 𝑄∗ is stable if Σ∗ is stable.4
Theorem 5.1. If a refinement satisfies Axioms A, B, and C then each solution is stable.
The proof occupies the remainder of this section.
5.1. Stable Sets of Equilibria. The proof begins by characterizing stable sets for games
in the class considered here. This characterization in Theorem 5.2 is somewhat simpler than
the general definition in Mertens [22] and for readers unfamiliar with homology theory it can
be taken as the definition.
Let Σ̄∗ be a component of the equilibria of Γ in terms of mixed strategies, and let Σ̄∗𝑛 be
the projection of Σ̄∗ in Σ𝑛 . Also let Σ∗ be a component of the undominated equilibria of the
game Γ that is contained in Σ̄∗ . Let 𝑄∗ be the image of Σ∗ under 𝜌 and for each 𝑛 let 𝑄∗𝑛
be the image of Σ∗𝑛 under 𝜌𝑛 , i.e. represented in enabling strategies.
By genericity, all equilibria in Σ̄∗ induce the same distribution over outcomes. Therefore,
for each node 𝑥, 𝛽(𝑥, 𝜎) is the same for all 𝜎 ∈ Σ̄∗ ; in particular, if 𝑥 belongs to information
set ℎ ∈ 𝐻𝑛 and ℎ is on an equilibrium path then 𝛽𝑛 (ℎ, 𝜎𝑛 ) is the same for every equilibrium
strategy 𝜎𝑛 of player 𝑛 in Σ̄∗ . We therefore denote these probabilities by 𝛽𝑛∗ (𝑥) and 𝛽𝑛∗ (ℎ).
Let 𝐻𝑛∗ be the collection of information sets ℎ ∈ 𝐻𝑛 of player 𝑛 such that 𝛽𝑛∗ (ℎ) > 0 and let
𝐴∗𝑛 be the set of actions at information sets in 𝐻𝑛∗ that are chosen with positive probability
¯ ∗ , where 𝐵
¯ ∗ is the set of profiles of behavioral strategies induced by
by the equilibria in 𝐵
equilibria in Σ̄∗ .
Let 𝑆𝑛0 ⊂ 𝑆𝑛 be the set of pure strategies 𝑠0𝑛 with the property that, at each information
set ℎ ∈ 𝐻𝑛∗ that 𝑠0𝑛 does not exclude, 𝑠0𝑛 prescribes an action in 𝐴∗𝑛 . Let 𝑆𝑛1 = 𝑆𝑛 ∖ 𝑆𝑛0 , i.e.
each pure strategy 𝑠𝑛 in 𝑆𝑛1 chooses a non-equilibrium action at some information set ℎ ∈ 𝐻𝑛∗
that it does not exclude.
For 𝑖 = 0, 1, let Σ𝑖𝑛 be the set of mixed strategies whose support is contained in 𝑆𝑛𝑖 . Observe
that the support of 𝑛’s strategy in every equilibrium in Σ̄∗ is contained in 𝑆𝑛0 and that every
4Alternatively, one can apply the definition of stability directly to 𝑄∗ as a component of equilibria,
represented in terms of enabling strategies, in the graph over the space of perturbations of players’ enabling
strategies as in footnote 1. As noted by Mertens [24, 25], more generally one can apply analogs of stability
and Axioms A, B, and C to games in strategic form for which each player’s strategy set is a convex polyhedron
and payoffs are multilinear functions defined on the product of players’ strategy sets.
AXIOMATIC EQUILIBRIUM SELECTION
11
strategy in 𝑆𝑛0 is a best reply against every equilibrium in Σ̄∗ . Thus Σ̄∗𝑛 is contained in Σ0𝑛
and Σ̄∗ = Σ̄∗1 × Σ̄∗2 . Hence Σ∗ = Σ∗1 × Σ∗2 where Σ∗𝑛 is a component of the intersection of Σ̄∗𝑛
with the set of undominated strategies.
¯ ∗ is completely mixed; by genericity,
If 𝑆𝑛1 is empty for each 𝑛 then each equilibrium in 𝐵
¯ ∗ is a singleton and its equivalent mixed strategy is stable. Thus if a solution concept
𝐵
satisfying our axioms selects this equilibrium, it is automatically stable. The only interesting
case, therefore, is one where 𝑆𝑛1 is nonempty for at least one of the players. In order to avoid
dealing with different cases, we assume that 𝑆𝑛1 is nonempty for each 𝑛. Along the way we
indicate how the proof changes when 𝑆𝑛1 is empty for exactly one player.
Let 𝑃𝑛0 be 𝑛’s set of enabling strategies in the image of Σ0𝑛 under 𝜌𝑛 . Let 𝑍𝑛1 ⊂ 𝑍 be the
set of terminal nodes 𝑧 such that (ℎ, 𝑎) ≺ 𝑧 for some ℎ ∈ 𝐻𝑛∗ and 𝑎 ∈
/ 𝐴∗𝑛 . Let 𝑍𝑛0 = 𝑍𝑛 ∖ 𝑍𝑛1 .
Then 𝑃𝑛0 is the set of 𝑝𝑛 ∈ 𝑃𝑛 such that 𝑝𝑛 (𝑧) = 0 for all 𝑧 ∈ 𝑍𝑛1 and thus 𝑃𝑛0 is a face
of 𝑃𝑛 . However, the image 𝑃𝑛1 of Σ1𝑛 under 𝜌𝑛 need not be a face of 𝑃𝑛 . For 𝑖 = 0, 1, let
𝑃 𝑖 = 𝑃1𝑖 × 𝑃2𝑖 and define ℙ = 𝑃 0 × 𝑃 1 .
𝑍1
For each enabling strategy 𝑝𝑛 ∈ 𝑃𝑛 , let Ψ𝑍𝑛1 (𝑝𝑛 ) be the projection of 𝑝𝑛 to ℝ+𝑛 ; then
Ψ𝑍𝑛1 (𝑝𝑛 ) = 0 iff 𝑝𝑛 ∈ 𝑃𝑛0 . Fix a point 𝑝¯𝑚 in the interior of 𝑃𝑚 and define 𝜂𝑛 : 𝑃𝑛 → ℝ by
∑
𝑝𝑚 (𝑧)𝑝𝑛 (𝑧), where 𝑝0 is Nature’s enabling strategy. Then 𝜂(𝑝𝑛 ) = 0 iff
𝜂(𝑝𝑛 ) = 𝑧∈𝑍𝑛1 𝑝0 (𝑧)¯
𝑝𝑛 ∈ 𝑃𝑛0 . Choose 𝜀 > 0 such that 𝜂𝑛 (𝑝𝑛 ) > 𝜀 for all 𝑝𝑛 ∈ 𝑃𝑛1 . Let ℋ𝑛 be the hyperplane
1
in ℝ𝑍𝑛 with normal (𝑝0 (𝑧)¯
𝑝𝑚 (𝑧))𝑧∈𝑍𝑛1 and constant 𝜀. Then ℋ𝑛 separates the origin from
Ψ𝑍𝑛1 (𝑃𝑛1 ). Let Π1𝑛 be the intersection of ℋ𝑛 with Ψ𝑍𝑛1 (𝑃𝑛 ). Let 𝜋
¯𝑛1 be the function from
𝑃𝑛 ∖ 𝑃𝑛0 to Π1𝑛 that maps each 𝑝𝑛 ∈
/ 𝑃𝑛0 to the point 𝜀(𝜂𝑛 (𝑝𝑛 ))−1 Ψ𝑍𝑛1 (𝑝𝑛 ).
In the following we invoke lexicographically optimal replies as defined in Blume, Brandenburger, and Dekel [1] and Govindan and Klumpp [3]. Recall that 𝑛’s strategy 𝜎𝑛 is
𝑘
lexicographically optimal against a sequence (𝜎𝑚
)𝑘=1,2,.. of 𝑚’s strategies if any alternative
𝑘
𝑗
strategy 𝜎
ˆ𝑛 that is a better reply to 𝜎𝑚 for some 𝑘 is a worse reply to 𝜎𝑚
for some 𝑗 < 𝑘.
∗
∗
0 1
1
∗
1
Given 𝑄 , let 𝒬 be the set of those (𝑞 , (𝑝 , 𝑝 ), 𝜋 ) ∈ 𝑄 × ℙ × Π such that there exist
0 0
𝑟 , 𝑝˜𝑛 ∈ 𝑃𝑛0 , 𝑟1 ∈ 𝑃𝑛1 , and for each 𝑛 scalars 𝜆0𝑛 , 𝜆1𝑛 , 𝜇1𝑛 in the interval (0, 1] such that, if
𝑞𝑛0 = 𝜆0𝑛 𝑝0𝑛 + (1 − 𝜆0𝑛 )𝑟𝑛0
and
𝑞𝑛1 = (1 − 𝜆1𝑛 )˜
𝑝0𝑛 + 𝜆1𝑛 (𝜇1𝑛 𝑝1𝑛 + (1 − 𝜇1𝑛 )𝑟𝑛1 ) ,
then for each 𝑛:
¯𝑛1 (𝑞𝑛1 ) = 𝜋𝑛1 .
(i) 𝜋
1
0
∗
).
, 𝑞𝑚
(ii) 𝑞𝑛0 , and 𝑟𝑛0 if 𝜆0𝑛 < 1, are lexicographically optimal replies against (𝑞𝑚
, 𝑞𝑚
∗
1
1
(iii) If 𝜇𝑛 < 1 then 𝑟𝑛 is an optimal reply against 𝑞𝑚 and lexicographically as good a reply
1
0
∗
) as other strategies in 𝑃𝑛1 .
, 𝑞𝑚
, 𝑞𝑚
against (𝑞𝑚
1
is not) then the set Π1𝑛 is empty so we set ℙ = 𝑃 0 × 𝑃𝑚1 and
In case 𝑆𝑛1 is empty (and 𝑆𝑚
1
), and we drop the optimality requirement
points in 𝒬 then have the form (𝑞 ∗ , (𝑝0 , 𝑝1𝑚 ), 𝜋𝑚
(iii) for 𝑛.
12
SRIHARI GOVINDAN AND ROBERT WILSON
The set 𝒬 is the graph of lexicographically optimal replies to possible deviations from
equilibria in 𝑄∗ . The formulation appears complicated only because of the need to consider
for each player 𝑛 both strategies 𝑝0𝑛 and 𝑝1𝑛 that do and do not adhere to equilibrium play,
and also possible mixtures of these with others having the same properties, so that altered
probabilities of actions are included. For each equilibrium 𝑞 ∗ one considers for each player 𝑛 a
pair (𝑝0𝑛 , 𝑝1𝑛 ) of enabling strategies such that 𝑝0𝑛 conforms to the equilibrium and 𝑝1𝑛 deviates.
Further, one considers a mixture 𝑞𝑛0 of 𝑝0𝑛 and some other conforming strategy 𝑟𝑛0 , and also
a mixture 𝑞𝑛1 of some conforming strategy 𝑝˜0𝑛 and a mixture of the nonconforming strategy
𝑝1𝑛 and some other nonconforming strategy 𝑟𝑛1 , where condition (i) requires that 𝑞𝑛1 yields
the specified probabilities 𝜋 1 on those terminal nodes excluded by equilibrium behavior.
From these strategies one obtains the sequence (𝑞𝑛∗ , 𝑞𝑛0 , 𝑞𝑛1 ) of alternative hypotheses about
𝑛’s strategies, ordered lexicographically. For the case that the mixtures give positive weight
to the alternative conforming strategy 𝑟𝑛0 and nonconforming strategy 𝑟𝑛1 , one requires in
(ii) that 𝑟𝑛0 is lexicographically optimal against the other’s sequence, and in (iii) that 𝑟𝑛1 is
an optimal reply to the equilibrium 𝑞 ∗ and lexicographically optimal among nonconforming
strategies. The point (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) is then in the graph above the point (𝑝0 , 𝑝1 ) describing
the players’ primary conforming and nonconforming strategies, if each player 𝑛’s mixture 𝑞𝑛0
of 𝑝0𝑛 and 𝑟𝑛0 is lexicographically optimal against the other’s sequence.
Let Ψ : 𝒬 → ℙ be the natural projection, i.e. Ψ(𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) = (𝑝0 , 𝑝1 ). Let ∂𝒬 =
Ψ−1 (∂ℙ). The projection map Ψ is essential if every continuous map 𝜙 : 𝒬 → ℙ has a point
of coincidence with Ψ, i.e. 𝜙(𝑥) = Ψ(𝑥) for some 𝑥 ∈ 𝒬. The following characterization of a
stable set is proved in Appendix C.
Theorem 5.2. (𝒬, ∂𝒬) is a pseudomanifold of the same dimension as (ℙ, ∂ℙ). Moreover,
𝑄∗ is stable if and only if the projection map Ψ : (𝒬, ∂𝒬) → (ℙ, ∂ℙ) is essential.
In spite of the desirable properties 1, 2, 3 listed in Section 1, the definition of stability
via the essentiality of the projection map has been a major impediment to justifying it
as an economically relevant refinement. For instance, it implies more than the intuitively
plausible requirement that the projection map should be surjective so that there exist nearby
equilibria of nearby games obtained by perturbing players’ strategy sets. However, the proof
that follows shows that for the class of games considered here its implications are precisely
the same as the conjunction of the decision-theoretic Axioms A, B, and C, each of which has
concrete economic significance.
5.2. Plan of the Proof. First we outline the method for proving Theorem 5.1. That a
stable set satisfies the axioms is shown by Mertens [22]. Here show that the axioms imply
that a solution is stable.
ˆ ≡ 𝜌(Σ̂) be the set of enabling strategies
Let Σ̂ ⊂ Σ be a solution of the game Γ and let 𝑄
equivalent to Σ̂. By Axiom A, Σ̂ is a connected set of equilibria in undominated strategies.
AXIOMATIC EQUILIBRIUM SELECTION
13
ˆ is
Hence it belongs to a component Σ∗ of equilibria in undominated strategies and thus 𝑄
ˆ equals 𝑄∗ and is stable. We accomplish
contained in 𝑄∗ ≡ 𝜌(Σ∗ ). We will show that 𝑄
this as follows. We take an arbitrary point 𝑞 0,∗ ∈ 𝑄∗ and an arbitrary neighborhood 𝑈 (𝑞 0,∗ )
of 𝑞 0,∗ . Then we construct a corresponding sequence of metagames Γ̃𝛿 as a parameter 𝛿
converges to zero. Using Axioms B and C, for each metagame Γ̃𝛿 in the sequence, there
ˆ Take
exists a quasi-perfect equilibrium whose projection, call it 𝑞 0,𝛿 , to 𝑃 is contained in 𝑄.
ˆ We will show that: (i) the
any such sequence of 𝑞 0,𝛿 converging to some point 𝑞 0,0 in 𝑄.
ˆ = 𝑄∗ ; and (ii) the existence of such a sequence
limit point 𝑞 0,0 belongs 𝑈 (𝑞 0,∗ ), hence 𝑄
converging to a limit point in 𝑈 (𝑞 0,∗ ) implies that the projection map Ψ is essential, hence
𝑄∗ is stable.
5.3. Preliminaries. In this subsection we lay the groundwork for the metagames to be
constructed in the proof.
ˆ let (𝒬, ∂𝒬) be the associated pseudomaniFor the given set 𝑄∗ containing the solution 𝑄,
fold, as constructed in subsection 5.1. Let 𝑞 0,∗ be an arbitrary point in 𝑄∗ and let 𝑈 (𝑞 0,∗ ) be
0
a neighborhood of 𝑞 0,∗ . For each player 𝑚, choose a point 𝑝0,∗
𝑚 in the interior of 𝑃𝑚 against
which 𝑞 0,∗ is a best reply and strategies in 𝑃𝑛1 are inferior replies. Such a choice is possible
by genericity of payoffs: the interior of the projection of Σ̄∗𝑚 , which is the component of 𝑚’s
equilibrium strategies that contains Σ∗𝑚 , belongs to the interior of 𝑃𝑚0 and all strategies in
𝑃𝑛1 are inferior replies against every such point. Since 𝑞𝑛0,∗ belongs to 𝑄∗𝑛 , which consists only
of undominated (hence admissible) strategies, there exists a point 𝑝𝑚 in the interior of 𝑃𝑚
against which 𝑞𝑛0,∗ is a best reply. 𝑝𝑚 is equivalent to a completely mixed strategy 𝜎𝑚 in Σ𝑚 .
Also, by the genericity of payoffs, we can choose 𝑝𝑚 to be such that strategies in 𝑃𝑛0 that do
not belong to the face containing 𝑞𝑛0,∗ in its interior are strictly inferior replies against 𝑝𝑚 .
0
1
𝑖
Express 𝜎𝑚 as a convex combination of 𝜎𝑚
and 𝜎𝑚
, where for 𝑖 = 0, 1, 𝜎𝑚
belongs to the
𝑖
0
1,∗
0
1
interior of Σ𝑚 . Let 𝑝𝑚 and 𝑝𝑚 be the enabling strategies that are equivalent to 𝜎𝑚
and 𝜎𝑚
,
0
1,∗
0
1
respectively. Then 𝑝𝑚 and 𝑝𝑚 are in the relative interiors of 𝑃𝑚 and 𝑃𝑚 respectively, and
∗
0,∗
0,∗ 1,∗
1,∗
𝑝𝑚 is a convex combination of 𝑝0𝑚 and 𝑝1,∗
𝑚 . It follows that 𝑥 ≡ (𝑞 , (𝑝 , 𝑝 ), 𝜋 ) belongs
1
to 𝒬 ∖ ∂𝒬, where for each 𝑛, 𝜋𝑛1,∗ = 𝜋
¯𝑛1 (𝑝𝑛1,∗ ), and in the definition of 𝒬, 𝑞𝑛0 is 𝑝0,∗
𝑛 , and 𝑞𝑛 is
𝑝𝑛 , which is a convex combination of 𝑝0𝑛 and 𝑝1,∗
𝑛 .
It follows from our construction in Appendix C that 𝑥∗ belongs to the interior of a polyhedron of the same dimension as ℙ. Therefore, we can choose a neighborhood 𝑉 (𝑥∗ ) of 𝑥∗
that is homeomorphic to a simplex, is contained in 𝒬 ∖ ∂𝒬, and is such that the projection
onto the first factor is contained in the neighborhood 𝑈 (𝑞 0,∗ ), i.e. if (𝑞 0 , (𝑝0 , 𝑝1 ), 𝜋 1 ) ∈ 𝑉 (𝑥∗ )
then 𝑞 0 ∈ 𝑈 (𝑞 0,∗ ).
In Appendix D we construct a continuous map 𝑔 : 𝒬 → ℙ and a constant 𝛼 > 0 such
that ∥𝑔(𝑥) − Ψ(𝑥)∥ ⩽ 𝛼 for some 𝑥 ∈ 𝒬 only if Ψ is essential and 𝑥 ∈ 𝑉 (𝑥∗ ). Now extend
the map 𝑔 to the whole of 𝑃 0 × ℙ × Π in an arbitrary fashion, calling it still 𝑔. Also, we
14
SRIHARI GOVINDAN AND ROBERT WILSON
will now view Ψ as the projection from 𝑃 0 × ℙ × Π to ℙ. Choose a triangulation 𝒦𝑛𝑖 of
𝑃𝑛𝑖 for each 𝑛 and 𝑖 = 0, 1 such that the diameter of each simplex is no more than 𝛼/2.
1
For each 𝑛 there exists for each 𝑖 a triangulation ℒ𝑖𝑛 of 𝑃𝑛𝑖 and a triangulation ℒΠ
𝑛 of Π𝑛
such that, letting ℒ be the resulting multisimplicial subdivision of 𝑃 0 × ℙ × Π1 , 𝑔 has a
multisimplicial approximation 𝑔˜ [8, Theorem 6] with the triangulation of the range given by
∏
𝒦 = 𝑛,𝑖 𝒦𝑛𝑖 . Observe that if for some 𝑥 = (𝑞 0 , (𝑝0 , 𝑝1 ), 𝜋 1 ) there exists a multisimplex 𝐾
that contains Ψ(𝑥) and 𝑔˜(𝑥), then ∥𝑔(𝑥) − Ψ(𝑥)∥ ⩽ 𝛼: this follows from the fact that, since 𝑔˜
is a multisimplicial approximation of 𝑔, 𝑔˜(𝑥) belongs to the multisimplex that contains 𝑔(𝑥)
in its interior. An important implication of this observation is that if, in particular, this 𝑥
also belongs to 𝒬, then Ψ is essential, 𝑥 ∈ 𝑉 (𝑥∗ ), and 𝑞 0 ∈ 𝑈 (𝑞 0,∗ ). Thus, that such a point
belongs to 𝒬 will be the final step of the proof in subsection 5.12.
As in [8, Appendix B], now take a further polyhedral subdivision 𝒯 of ℒ and let 𝛾 be
the convex function generated by 𝒯 , i.e. 𝛾 is piecewise linear and linear precisely on each
full-dimensional polyhedron of the subdivision.
5.4. A Game with Redundant Strategies. In this subsection we construct from Γ a
larger game by adding redundant pure strategies that will be the basis for the metagame
specified in subsection 5.5. Because Axiom C implies Invariance, the solutions of Γ are
equivalent to the solutions of this larger game.
For each fixed 𝑝ˆ = (𝑝0 , 𝑝1 ) ∈ ℙ and 𝛿 ∈ (0, 1), consider the following game Γ(𝛿, 𝑝ˆ). Each
player 𝑛 chooses a strategy in the following manner (and unaware of his opponent’s choices).
Initially, player 𝑛 provisionally chooses a pure strategy 𝑠0𝑛 ∈ 𝑆𝑛0 , or he rejects all strategies
in 𝑆𝑛0 .
∙ If initially he chooses a strategy 𝑠0𝑛 then at a subsequent second stage he can retain
𝑠0𝑛 or revise his choice. If he chooses to revise his choice, then at a third stage the
revisions available are ‘duplicate’ pure strategies in the set 𝑇𝑛0 (𝛿, 𝑝0𝑛 ) consisting of all
mixed strategies of the form 𝑡𝑛 (𝛿, 𝑝0𝑛 ) ≡ (1 − 𝛿)𝑡𝑛 + 𝛿𝑝0𝑛 for some 𝑡𝑛 ∈ 𝑆𝑛0 .
∙ If he rejects all strategies in 𝑆𝑛0 at the first stage, then at a second stage he can
choose among the strategies in 𝑇𝑛0 (𝛿, 𝑝0𝑛 ) or reject them all.5 If he rejects them all
then at a third stage he chooses among the pure strategies in 𝑆𝑛1 ∪ 𝑇𝑛1 (𝛿, 𝑝1𝑛 ), where
each strategy in 𝑇𝑛1 (𝛿, 𝑝1𝑛 ) is a duplicate of the form 𝑡𝑛 (𝛿, 𝑝1𝑛 ) ≡ (1 − 𝛿)𝑡𝑛 + 𝛿𝑝1𝑛 for
some 𝑡𝑛 ∈ 𝑆𝑛0 .
In Γ(𝛿, 𝑝ˆ) the set of 𝑛’s pure strategies is 𝑆˜𝑛 (𝛿, 𝑝ˆ𝑛 ) ≡ 𝑆𝑛 ∪ 𝑇𝑛0 (𝛿, 𝑝0𝑛 ) ∪ 𝑇𝑛1 (𝛿, 𝑝1𝑛 ). Thus, game
Γ(𝛿, 𝑝ˆ) has the same reduced normal form as Γ.
5It
would have sufficed, at this stage, to give player 𝑛 the option of playing just the strategy 𝑝0𝑛 instead
of all the strategies in 𝑇𝑛0 (𝛿, 𝑝0𝑛 ), which we do only for economy in notation.
AXIOMATIC EQUILIBRIUM SELECTION
15
5.5. Extensive Form of the Metagames. Now we specify a family of similar metagames
Γ̃𝛿 , one for each 𝛿 ∈ (0, 1).
Before the insiders play, thirteen outsiders, denoted players 𝑜0 and 𝑜𝑖𝑛,𝑗 for 𝑛 = 1, 2, 𝑖 = 0, 1
and 𝑗 = 1, 2, 3, move simultaneously. Outsider 𝑜0 chooses a full-dimensional polyhedron 𝑇
of the polyhedral complex 𝒯 . Outsider 𝑜𝑖𝑛,𝑗 , for 𝑛 = 1, 2, 𝑗 = 1, 3 and 𝑖 = 0, 1, chooses a
point in the vertex set 𝑊𝑛𝑖 of 𝒦𝑛𝑖 . Outsider 𝑜0𝑛,2 chooses a point in a finite subset 𝑆𝑛0,𝛿 of 𝑃𝑛0
chosen such that every point in 𝑃𝑛0 is within 𝛿 of some point in 𝑆𝑛0,𝛿 ; outsider 𝑜1𝑛,2 chooses a
point in a finite subset 𝑆𝑛1,𝛿 of Π1𝑛 such that every point in Π1𝑛 is within 𝛿 of some point in
𝑆𝑛1,𝛿 .
For outsider 𝑜𝑖𝑛,1 , each pure strategy 𝑣𝑛𝑖 corresponds to a point in 𝑃𝑛𝑖 denoted 𝑝𝑖𝑛 (𝑣𝑛𝑖 ).
𝑖
𝑖
Therefore, each mixed strategy 𝜎𝑛,1
corresponds to a point in 𝑃𝑛𝑖 , denoted 𝑝𝑖𝑛 (𝜎𝑛,1
), which is
obtained by taking the appropriate average of the points induced by the pure strategies in
𝑖
0
0
the support of 𝜎𝑛,1
. Likewise a mixed strategy 𝜎𝑛,2
of 𝑜0𝑛,2 corresponds to a point 𝑞𝑛0 (𝜎𝑛,2
) in
1
1
) in Π1𝑛 .
𝑃𝑛0 , and a mixed strategy 𝜎𝑛,2
of 𝑜1𝑛,2 corresponds to a point 𝜋𝑛1 (𝜎𝑛,2
A mixed-strategy profile 𝜎
˜𝑜 for the outsiders induces a point (𝑞 0 (˜
𝜎𝑜 ), (𝑝0 (˜
𝜎𝑜 ), 𝑝1 (˜
𝜎𝑜 )), 𝜋 1 (˜
𝜎𝑜 ))
0
1
𝑖
𝑖
0
in 𝑃 × ℙ × Π , where for each 𝑛 and 𝑖, 𝑝𝑛 (˜
𝜎𝑜 ) depends on the choice by 𝑜𝑛,1 , and 𝑞𝑛 (˜
𝜎𝑜 ) and
1
0
1
𝜋
¯𝑛 (˜
𝜎𝑜 ) depend on the choices by 𝑜𝑛,2 and 𝑜𝑛,2 respectively.
After each pure-strategy profile 𝑠˜𝑜 of the outsiders there follows a copy of the game
Γ(𝛿, 𝑝ˆ(˜
𝑠𝑜 )). That is, if in the profile 𝑠˜𝑜 outsiders 𝑜𝑖𝑛,1 choose points 𝑣𝑛𝑖 , then there follows
a copy of Γ(𝛿, 𝑝ˆ) after these choices, where for each 𝑛, 𝑝ˆ𝑛 = (𝑝0𝑛 (𝑣𝑛0 ), 𝑝1𝑛 (𝑣𝑛1 )). However, the
information sets in Γ̃𝛿 are such that the insiders play without knowing which copy of Γ(𝛿, 𝑝ˆ)
they are playing. The sets of duplicate strategies available are therefore now denoted by
𝑇𝑛0 (𝛿) and 𝑇𝑛1 (𝛿), omitting the reference to 𝑝0𝑛 and 𝑝1𝑛 , since the insiders are uninformed
about which mixtures were implemented by outsiders. Put differently, for 𝑖 = 0, 1 and
𝑡𝑛 ∈ 𝑆𝑛0 , the exact duplicate strategy implemented by choosing 𝑡𝑖𝑛 (𝛿) ∈ 𝑇𝑛𝑖 (𝛿) depends on the
choice by outsider 𝑜𝑖𝑛,1 which insiders do not observe. Thus in the metagame Γ̃𝛿 , player 𝑛’s
set of pure strategies, up to duplication of pure strategies, is 𝑆˜𝑛 (𝛿) ≡ 𝑆𝑛 ∪ 𝑇𝑛0 (𝛿) ∪ 𝑇𝑛1 (𝛿).
That the metagame Γ̃𝛿 embeds Γ follows from Proposition 3.3. The strategies in 𝑆𝑛 are
available as pure strategies in 𝑆˜𝑛 (𝛿) and the other pure strategies, which belong to 𝑇𝑛𝑖 (𝛿),
for 𝑖 = 0, 1, implement mixtures in Σ𝑛 that depend on the choices of outsiders 𝑜𝑖𝑛,1 .
5.6. Outsiders’ Payoffs in the Metagames. Next we describe the outsiders’ payoffs in
each metagame Γ̃𝛿 .
The payoffs to 𝑜0 depend on the choices of all outsiders except outsiders 𝑜𝑖𝑛,3 for 𝑖 = 0, 1 and
𝑛 = 1, 2. Recall that the convex function 𝛾 is linear over each full-dimensional polyhedron 𝑇
of 𝒯 . This linear function extends uniquely to a linear function 𝛾𝑇 over 𝑃 0 × ℙ × Π1 . Every
16
SRIHARI GOVINDAN AND ROBERT WILSON
mixed strategy profile of the other outsiders induces a unique point (˜
𝑞 , 𝑝ˆ, 𝜋 1 ) ∈ 𝑃 0 × ℙ × Π1
and 𝑜0 ’s payoff from choosing 𝑇 is 𝛾𝑇 (˜
𝑞 , 𝑝ˆ, 𝜋 1 ).
Outsider 𝑜𝑖𝑛,1 wants to mimic 𝑜𝑖𝑛,3 . In particular, if he chooses 𝑣𝑛𝑖 and 𝑜𝑖𝑛,3 chooses 𝑤𝑛𝑖 then
his payoff is one if 𝑣𝑛𝑖 = 𝑤𝑛𝑖 and zero otherwise.
Outsider 𝑜0𝑛,2 wants to mimic the actual choice implemented by player 𝑛 when this choice
belongs to 𝑃𝑛0 . Similarly, outsider 𝑜1𝑛,2 wants to mimic the 𝜋𝑛1 implied by 𝑛’s choice when he
𝑖
plays a strategy in 𝑃𝑛1 . Specifically, for 𝑖 = 0, 1, let 𝜑𝑖𝑛 : ℝ𝑍𝑛 → ℝ be the function given by
∑
𝑖
𝜑𝑖𝑛 (𝑟) = 𝑧∈𝑍𝑛𝑖 𝑟𝑧2 . For each 𝑟 ∈ ℝ𝑍𝑛 , let 𝜉𝑛𝑖 (𝑟, ⋅) be the affine approximation to 𝜑𝑖𝑛 at 𝑟, i.e.
∑
𝑖
for each 𝑟′ ∈ ℝ𝑍𝑛 , 𝜉𝑛𝑖 (𝑟, 𝑟′ ) = 𝑧 (𝑟𝑧2 + 2𝑟𝑧 (𝑟𝑧′ − 𝑟𝑧 )). Suppose now that 𝑜𝑖𝑛,2 chooses a pure
strategy 𝑠𝑖,𝛿
˜𝑛 in 𝑆˜𝑛 (𝛿). If 𝑠˜𝑛 is in 𝑇𝑛0 (𝛿) or 𝑇𝑛1 (𝛿) then let 𝑞𝑛
𝑛,2 and 𝑛 chooses a pure strategy 𝑠
be the actual strategy in 𝑃𝑛 that is implemented based on 𝑝0𝑛 (𝑣𝑛0 ) or 𝑝1𝑛 (𝑣𝑛1 ) where for each
𝑖, 𝑣𝑛𝑖 is the choice of outsider 𝑜𝑖𝑛,1 ; and otherwise let 𝑞𝑛 = 𝑠˜𝑛 . For outsider 𝑜0𝑛,2 , his payoff is
1
0
zero if 𝑞𝑛 ∈
/ 𝑃𝑛0 ; otherwise, it is 𝜉𝑛0 (𝑠0,𝛿
𝑛,2 , 𝑞𝑛 ). For outsider 𝑜𝑛,2 , his payoff is zero if 𝑞𝑛 ∈ 𝑃𝑛 ;
otherwise it is 𝜉𝑛1 (𝑠1,𝛿
¯𝑛1 (𝑞𝑛 )).
𝑛,2 , 𝜋
The payoff to outsider 𝑜𝑖𝑛,3 depends on the choices of all other outsiders. If 𝑜0 chooses
a polyhedron 𝑇 then there exists a unique multisimplex 𝐿 of ℒ that contains 𝑇 . For each
vertex 𝑤𝑛𝑖 of 𝑊𝑛𝑖 , and each vertex 𝑣˜ of 𝐿, let 𝑢𝑖𝑛,2 (𝑇, 𝑣˜, 𝑤𝑛𝑖 ) = 1 if 𝑤𝑛𝑖 is the image of 𝑣˜ under 𝑔˜𝑛𝑖
and zero otherwise, where 𝑔˜𝑛𝑖 is the (𝑛, 𝑖)-th coordinate map of 𝑔˜. The function 𝑢𝑖𝑛,2 extends
multilinearly to 𝐿 and, since 𝐿 is full-dimensional, to the whole of 𝑃 0 × ℙ × Π1 , denoted
still by 𝑢𝑖𝑛,2 (𝑇, ⋅, 𝑤𝑛𝑖 ). Given an arbitrary mixed strategy of the other players, if 𝑜0 chooses
𝑇 and 𝑜𝑖𝑛,2 chooses 𝑤𝑛𝑖 then the payoff of 𝑜𝑖𝑛,2 is 𝑢𝑖𝑛,2 (𝑇, (𝑝, 𝑞), 𝑤𝑛𝑖 ), where (𝑝, 𝑞) is the point
in 𝑃 0 × ℙ × Π1 induced by the mixed strategies of the other players.
5.7. Outsiders’ Strategies in a Quasi-Perfect Equilibrium. In this subsection we derive the relevant features of outsiders’ strategies in a quasi-perfect equilibrium.
By Axioms B and C, in the metagame Γ̃𝛿 there exists a quasi-perfect equilibrium ˜𝑏𝛿 whose
equivalent mixed-strategy profile 𝜎
˜ 𝛿 belongs to a solution and whose image under the map
ˆ for the original game Γ.
from the metagame Γ̃𝛿 to Γ is a point in the solution 𝑄
Each player 𝑛’s strategy in ˜𝑏𝛿 necessarily has the following feature. He avoids going to his
information set where his choices are among the strategies in 𝑆𝑛1 (𝛿) ∪ 𝑇𝑛1 (𝛿), since each of
these strategies chooses a non-equilibrium action at some information set on the equilibrium
path. Let 𝑞𝑛0,𝛿 be 𝑛’s actual strategy in 𝑃𝑛0 that is implemented by 𝑛’s strategy in the profile
˜𝑏𝛿 in the metagame Γ̃𝛿 . By construction, 𝑞 0,𝛿 belongs to 𝑄.
ˆ
𝑛
𝛿
0,𝛿
0,𝛿 1,𝛿
1,𝛿
0
Let 𝑥˜ ≡ (˜
𝑞 , (𝑝 , 𝑝 ), 𝜋
˜ ) be the point in 𝑃 × ℙ × Π1 that is induced by the profile
˜𝛿.
𝜎
˜𝑜𝛿 of the outsiders’ strategies in the equilibrium 𝜎
Under ˜𝑏𝛿 , after 𝑛 has rejected all strategies in 𝑆𝑛0 and 𝑇𝑛0 (𝛿), consider the strategy implemented by 𝑛. Let (1 − 𝛼𝑛1,𝛿 ) be the total probability of choosing a duplicate in 𝑇𝑛1 (𝛿) under
AXIOMATIC EQUILIBRIUM SELECTION
17
˜𝑏𝛿 . Then 𝑛’s choice at this information set is equivalent to an enabling strategy in 𝑃𝑛 of the
form
1,𝛿
1,𝛿 1,𝛿
𝑞ˇ𝑛𝛿 ≡ (1 − 𝛼𝑛1,𝛿 )((1 − 𝛿)ˇ
𝑝0,𝛿
𝑛 + 𝛿𝑝𝑛 ) + 𝛼𝑛 𝑟𝑛 ,
where: (i) 𝑝ˇ0𝑛 (𝛿) is the mixture over strategies 𝑡𝑛 such that the strategy 𝑡1𝑛 (𝛿) is played
1
1
with positive probability at this information set; (ii) 𝑝1,𝛿
𝜎𝑛,1
) is the enabling strategy
𝑛 = 𝑝𝑛 (˜
1
induced by the equilibrium strategy 𝜎
˜𝑛,1
of outsider 𝑜1𝑛,1 ; (iii) 𝑟𝑛1,𝛿 is the enabling strategy in
𝑃𝑛1 that is obtained from 𝑛’s actual mixture over strategies in 𝑆𝑛1 if 𝛼𝑛1,𝛿 > 0, and is arbitrary
otherwise. Let
−1
1,𝛿 1,𝛿
𝑞 1,𝛿 = (𝛿(1 − 𝛼𝑛1,𝛿 ) + 𝛼𝑛1,𝛿 ) (𝛿(1 − 𝛼𝑛1,𝛿 )𝑝1,𝛿
𝑛 + 𝛼𝑛 𝑟𝑛 ) .
¯𝑛1 (𝑞𝑛1,𝛿 ) ≡ 𝜋𝑛1,𝛿 .
Then 𝜋
¯𝑛1 (ˇ
𝑞𝑛𝛿 ) = 𝜋
The following lemma characterizes the important aspects of the outsiders’ equilibrium
strategies.
Lemma 5.3. The equilibrium strategies of the outsiders satisfy the following properties.
(1) For each 𝑛, suppose the vertices in the support 𝑊𝑛𝑖,𝛿 of 𝑜𝑖𝑛,3 ’s equilibrium strategy
span a simplex 𝐾𝑛𝑖,𝛿 of 𝒦𝑛𝑖 . Then 𝑝𝑛𝑖,𝛿 belongs to 𝐾𝑛𝑖,𝛿 .
(2) If every polyhedron in the support of 𝑜0 ’s strategy contains 𝑥˜𝛿 then, for each 𝑛 and
𝑖, the vertices in 𝑊𝑛𝑖,𝛿 span a simplex 𝐾𝑛𝑖,𝛿 , and 𝑔˜𝑛,𝑖 (˜
𝑥𝛿 ) belongs to the interior of a
¯ 𝑖,𝛿 that has 𝐾 𝑖,𝛿 as a face.
simplex 𝐾
𝑛
𝑛
(3) Every polyhedron in the support of 𝑜0 ’s strategy contains 𝑥˜𝛿 .
(4) 𝑞˜𝑛0,𝛿 is within 𝛿 of 𝑞𝑛0,𝛿 and 𝜋
˜𝑛1,𝛿 is within 𝛿 of 𝜋𝑛1,𝛿 .
Proof of Lemma. Outsider 𝑜𝑖𝑛,1 wants to mimic outsider 𝑜𝑖𝑛,3 . So, if the vertices of 𝑊𝑛𝑖,𝛿 span
a simplex 𝐾𝑛𝑖,𝛿 then the payoff to 𝑜𝑖𝑛,1 from choosing a vertex 𝑤𝑛𝑖 is positive if it belongs to
𝑊𝑛𝑖,𝛿, and zero otherwise. Point (1) follows.
¯ = 𝐿
˜ 0 × (𝐿0 × 𝐿1 ) × 𝐿Π be the unique multisimplex of ℒ that contains 𝑥˜𝛿 in its
Let 𝐿
interior. For each polyhedron 𝑇 in the support of 𝑜0 ’s strategy, there exists a full-dimensional
ˆ of ℒ that contains 𝑇 . Obviously 𝐿
ˆ has 𝐿
¯ as a face. 𝑜𝑖 ’s payoff from choosing
multisimplex 𝐿
𝑛,3
a strategy 𝑤𝑛𝑖 if 𝑜0 chooses such a 𝑇 , and given the strategies of the other outsiders, is positive
¯ under 𝑔˜𝑖 and zero otherwise. Since the image of the vertices
if it is the image of a vertex of 𝐿
𝑛
¯ 𝑛𝑖,𝛿 and the vertices of
¯ 𝑛𝑖,𝛿 , 𝑔˜𝑛𝑖 (˜
¯ under the coordinate function 𝑔˜𝑛𝑖 span a simplex 𝐾
𝑥𝛿 ) ∈ 𝐾
of 𝐿
¯ 𝑖,𝛿 . Therefore, point (2) follows.
𝑊𝑛𝑖,𝛿 span a face of 𝐾
𝑛
For each polyhedron 𝑇 of 𝒯 , 𝑜0 ’s payoff from 𝑇 is 𝛾𝑇 (˜
𝑥𝛿 ) and by construction, 𝛾𝑇 (˜
𝑥𝛿 ) ⩽
𝛾(˜
𝑥𝛿 ) with the inequality being strict iff 𝑥˜𝛿 does not belong to 𝑇 , which proves (3).
It remains to prove (4). The actual strategy implemented by 𝑛 is 𝑞𝑛0,𝛿 , which belongs to
𝑃𝑛0 . 𝑜0𝑛,2 ’s payoff function is such that his best replies to 𝑞𝑛0,𝛿 are the points in 𝑆𝑛0,𝛿 that are
closest to 𝑞𝑛0,𝛿 . Thus 𝑞˜𝑛0,𝛿 is within 𝛿 of 𝑞𝑛0,𝛿 . Since 𝑞𝑛0,𝛿 belongs to 𝑃𝑛0 , all of 𝑜1𝑛,2 ’s strategies
18
SRIHARI GOVINDAN AND ROBERT WILSON
yield a payoff of zero against 𝑞𝑛0,𝛿 . However, since the behavioral strategy ˜𝑏𝛿 is a quasi-perfect
equilibrium, there exists a sequence of completely mixed behavioral strategies ˜𝑏𝜀,𝛿 converging
𝛿
to ˜𝑏𝛿 against which 𝑜1𝑛,2 ’s equilibrium strategy 𝜎
˜𝑛,2
is a best reply. Under the sequence ˜𝑏𝜀,𝛿 ,
there is a positive probability that player 𝑛 rejects the strategies in 𝑆𝑛0 ∪ 𝑇𝑛0 (𝛿) and makes
𝛿
a choice among strategies in 𝑆𝑛1 ∪ 𝑇𝑛1 (𝛿). The fact that 𝜎
˜𝑛,2
is optimal against the sequence
𝛿
implies that it is optimal against the limiting choice 𝑞ˇ𝑛 there. Since 𝜋
¯𝑛1 (ˇ
𝑞𝑛𝛿 ) = 𝜋𝑛1,𝛿 , 𝑜1𝑛,2 ’s best
□
˜𝑛1,𝛿 is within 𝛿 of 𝜋𝑛1,𝛿 as well.
replies are within 𝛿 of 𝜋𝑛1,𝛿 and thus 𝜋
5.8. Limits of the Quasi-Perfect Equilibria of the Metagames. In this subsection we
derive the limits of the metagames’ quasi-perfect equilibria as 𝛿 ↓ 0.
Consider a sequence of 𝛿’s converging to zero and a corresponding sequence ˜𝑏𝛿 of quasiperfect equilibria in solutions of the metagames Γ̃𝛿 . Let 𝜎
˜ 𝛿 be an equivalent sequence of
mixed strategies and let (˜
𝑞 0,𝛿 , (𝑝0,𝛿 , 𝑝1,𝛿 ), 𝜋
˜ 1,𝛿 ) be the sequence in 𝑃 0 × ℙ × Π1 induced by the
outsiders’ strategies.
Let 𝑞˜0,0 , (𝑝0,0 , 𝑝1,0 ), and 𝜋
˜ 1,0 be the corresponding limits of 𝑞˜0,𝛿 , (𝑝0,𝛿 , 𝑝1,𝛿 ), and 𝜋
˜ 1,𝛿 . Let
ˆ By properties (1)-(3) of
𝑞 0,0 and 𝜋 1,0 be the limits of 𝑞 0,𝛿 and 𝜋 1,𝛿 . 𝑞 0,0 belongs to 𝑄.
𝑖,𝛿
¯ of 𝒦𝑖 that contains both
the previous lemma, for each 𝛿, 𝑛, 𝑖, there exists a simplex 𝐾
𝑛
𝑛
𝑔˜𝑛,𝑖 (˜
𝑞 0,𝛿 , (𝑝0,𝛿 , 𝑝1,𝛿 ), 𝜋
˜ 1,𝛿 ) and 𝑝𝑖,𝛿
,
with
the
former
belonging
to
its
interior.
By property (4)
𝑛
0,0
0,0
1,0
1,0
of the previous lemma, 𝑞˜ = 𝑞 and 𝜋
˜ =𝜋 .
¯ of ℒ and
By passing to a subsequence, we can assume that there exist multisimplices 𝐿
0,𝛿
0,𝛿
1,𝛿
1,𝛿
¯ of 𝒦 such that for all 𝛿, (˜
¯ and its image
𝐾
𝑞 , (𝑝 , 𝑝 ), 𝜋
˜ ) belongs to the interior of 𝐿
0,𝛿
1,𝛿
¯
¯ also contains (𝑝 , 𝑝 ). Going to the limit,
under 𝑔˜ belongs to the interior of 𝐾—hence
𝐾
¯ and its image under 𝑔˜ belongs to 𝐾;
¯ also (𝑝0,0 , 𝑝1,0 )
𝑥0 ≡ (𝑞 0,0 , (𝑝0,0 , 𝑝1,0 ), 𝜋 1,0 ) belongs to 𝐿
¯
belongs to 𝐾.
If we can show that 𝑥0 ∈ 𝒬, then, by construction, Ψ is essential and 𝑥0 ∈ 𝑉 (𝑥∗ ); therefore
𝑞 0,0 ∈ 𝑈 (𝑞 0,∗ ) and 𝑄∗ is stable, which proves the theorem. To show that 𝑥0 belongs to 𝒬,
it suffices to prove that 𝑥𝛿 ≡ (𝑞 0,0 , (𝑝0,𝛿 , 𝑝1,0 ), 𝜋 1,0 ) belongs to 𝒬 for all small 𝛿, since 𝒬 is
closed. The remainder of the proof establishes this property.
5.9. Insiders’ Strategies in a Quasi-Perfect Equilibrium. Next we derive the important features of the insiders’ strategies in quasi-perfect equilibria of the metagames, and their
limits as 𝛿 ↓ 0.
Let ˜𝑏𝜀,𝛿 be a sequence of completely mixed behavioral strategies converging to ˜𝑏𝛿 against
which for each insider 𝑛 and each information set of 𝑛 in Γ̃𝛿 , his continuation strategy as
given by 𝑏𝛿 is optimal. If 𝑛 chooses 𝑠𝑛 ∈ 𝑆𝑛0 in the first stage then in the second stage he has
the option of revising this strategy to play something in 𝑇𝑛0 (𝛿). Therefore, quasi-perfection
implies that player 𝑛 will end up implementing 𝑠𝑛 with positive probability in ˜𝑏𝛿𝑛 only if
this strategy is at least as good a reply against the sequence ˜𝑏𝜀,𝛿 as the strategies in 𝑇𝑛0 (𝛿).
AXIOMATIC EQUILIBRIUM SELECTION
19
Likewise, player 𝑛 has the option of playing a strategy in 𝑇𝑛0 (𝛿) before he decides to play a
strategy in 𝑆𝑛1 and even when he makes a choice among these strategies, he has the option
of choosing a strategy in 𝑇𝑛1 (𝛿). Therefore, at the information set that follows his choice of
avoiding strategies in 𝑆𝑛0 , ˜𝑏𝛿𝑛 assigns a positive probability to moving on to a third stage and
then choosing a strategy in 𝑇𝑛1 (𝛿) ∪ 𝑆𝑛1 only if one of these strategies is at least as good a
reply against the sequence ˜𝑏𝜀,𝛿 as all the strategies in 𝑇𝑛0 (𝛿). Furthermore, at the information
set obtained after 𝑛 avoids strategies in 𝑆𝑛0 ∪ 𝑇𝑛0 (𝛿), ˜𝑏𝛿𝑛 assigns a positive probability to a
strategy 𝑠𝑛 in 𝑆𝑛1 only if 𝑠𝑛 is at least as good a reply against the sequence ˜𝑏𝜀,𝛿 as all the
strategies in 𝑆𝑛1 ∪ 𝑇𝑛1 (𝛿).
Let 𝜎
˜ 𝜀,𝛿 be a sequence of mixed-strategy profiles in Γ̃𝛿 that is equivalent to the sequence
˜𝑏𝜀,𝛿 of behavioral-strategy profiles. For each player 𝑛, his strategy 𝜎
˜𝑛𝜀,𝛿 in the sequence is a
mixture over his pure strategy set 𝑆˜𝑛 = 𝑆𝑛 ∪ 𝑇𝑛0 (𝛿) ∪ 𝑇𝑛1 (𝛿). However, the implications of 𝑛’s
strategy (for 𝑚’s choices) depend on the choices of the outsiders through their implications
for strategies in 𝑇𝑛0 (𝛿) and 𝑇𝑛1 (𝛿). Each strategy 𝑡𝑖𝑛 (𝛿) plays 𝑡𝑛 with probability (1 − 𝛿) and
with probability 𝛿 plays a strategy in 𝑃𝑛𝑖 that is determined by 𝑜𝑖𝑛,1 ’s strategy. In order to
fully capture the impact that 𝑜𝑖𝑛,1 has on 𝑡𝑖𝑛 (𝛿), let 𝑇¯𝑛𝑖 (𝛿) be the union over all 𝑤𝑛𝑖 ∈ 𝑊𝑛𝑖 of
the sets 𝑇𝑛𝑖 (𝛿, 𝑝𝑖𝑛 (𝑤𝑛𝑖 )). Let 𝑆¯𝑛 = 𝑆𝑛 ∪ 𝑇¯𝑛0 (𝛿) ∪ 𝑇¯𝑛1 (𝛿) and let Σ̄𝑛 be the set of mixtures over
𝑆¯𝑛 .
The sequence 𝜎
˜ 𝜀,𝛿 induces a mixed strategy 𝜎
¯𝑛𝜀,𝛿 in 𝑆¯𝑛 for each 𝑛 as follows. For each
𝑠𝑛 ∈ 𝑆𝑛 , the probability 𝜎
¯𝑛𝜀,𝛿 (𝑠𝑛 ) of 𝑠𝑛 is 𝜎
˜𝑛𝜀,𝛿 (𝑠𝑛 ); for each 𝑖 = 0, 1, 𝑤𝑛𝑖 ∈ 𝑊𝑛𝑖 and 𝑡𝑛 ∈ 𝑆𝑛0 ,
𝑖
the probability 𝜎
¯𝑛𝜀,𝛿 (𝑡𝑖𝑛 (𝛿, 𝑝𝑖𝑛 (𝑤𝑛𝑖 ))) is 𝜎
˜𝑛𝜀,𝛿 (𝑡𝑖𝑛 (𝛿))˜
𝜎𝑜𝜀,𝛿
𝑖 (𝑤𝑛 ). From player 𝑚’s perspective it is
𝑛,1
the sequence 𝜎
¯𝑛𝜀,𝛿 , or rather its equivalent sequence in 𝑃𝑛 , that matters for his choice.
5.10. The Induced Lexicographic Probability System. The next step uses these sequences to obtain a representation of the insiders’ strategies as a lexicographic probability
system.
By Blume, Brandenburger, and Dekel [1, Appendix Proposition 2], we can construct for
𝑙 (𝛿),𝛿
each player 𝑛 a lexicographic probability system (LPS) Λ̄𝛿𝑛 = (¯
𝜎𝑛0,𝛿 , . . . , 𝜎
¯𝑛𝑛
strategies in 𝑆¯𝑛 such that for each 𝜀 in a subsequence converging to zero,
) over his
𝜎
¯𝑛𝜀,𝛿 = (1 − 𝜈0 (𝜀))(¯
𝜎𝑛0,𝛿 + 𝜈0 (𝑒)((1 − 𝜈1 (𝜀))¯
𝜎𝑛1,𝛿 + 𝜈1 (𝜀)((1 − 𝜈2 (𝜀))¯
𝜎𝑛1,𝛿 + ⋅ ⋅ ⋅ + 𝜈𝑙𝑛 (𝛿)−1 (𝜀)¯
𝜎𝑛𝑙𝑛 (𝛿),𝛿 )) ,
𝑙 (𝛿)
𝑛
where (𝜈0 (𝜀), . . . , 𝜈𝑙𝑛 (𝛿)−1 (𝜀)) is a sequence in ℝ++
converging to the origin. Moreover, 𝑙𝑛 (𝛿)
¯
depends only on the cardinality of 𝑆𝑛 (𝛿), which is independent of 𝛿. Let 𝑙𝑛0,𝛿 be the first level
in Λ̄𝛿𝑛 at which some strategy in 𝑇¯𝑛0 (𝛿) appears with positive probability. Let 𝑙𝑛1,𝛿 be the first
level of Λ̄𝛿𝑛 at which some strategy in 𝑆𝑛1 ∪ 𝑇¯𝑛1 (𝛿) appears with positive probability.
From the LPS Λ̄𝛿𝑛 construct an LPS Λ𝛿 = (𝑞𝑛0,𝛿 , . . . , 𝑞𝑛𝑙𝑛 ,𝛿 ) for Γ where for each 𝑙, 𝑞𝑛𝑙,𝛿 is an
enabling strategy in Γ that is equivalent to 𝜎
¯𝑛𝑙,𝛿 . 𝑞𝑛0,𝛿 is a lexicographic best reply against
20
SRIHARI GOVINDAN AND ROBERT WILSON
¯ 𝑖,𝛿 be the total probability of the strategies in 𝑇¯𝑖 (𝛿) under 𝜎
Λ𝛿𝑚 . If we let 𝜆
¯𝑛𝑖,𝛿 , then for each
𝑛
𝑛
𝑖,𝛿
¯ 𝑖,𝛿 𝜎
𝑤𝑖 ∈ 𝑊 𝑖 , the total probability under 𝜎
¯ 𝑙𝑛 of the strategies in 𝑇 𝑖 (𝛿, 𝑝𝑖 (𝑤𝑖 )) is 𝜆
˜ 𝛿𝑖 (𝑤𝑖 ),
𝑛
where
𝑛
𝑛
𝑛
𝜎𝑜𝛿𝑖 (𝑤𝑛𝑖 )
𝑛,1
is the probability assigned to vertex
𝑤𝑛𝑖
by outsider
𝑛
𝑜𝑖𝑛,1 ’s
𝑛
𝑛
𝑛
𝑜𝑛,1
equilibrium strategy
𝑖,𝛿
𝜎𝑜𝛿𝑖 . Since 𝑝𝑖𝑛 (˜
𝜎𝑜𝛿𝑖 ) is, by definition, 𝑝𝑖,𝛿
𝑛 , we have that level 𝑙𝑛 is expressible as a convex
𝑛,1
𝑛,1
𝑖 ,𝛿
𝑖 ,𝛿
𝑖 ,𝛿
𝑙𝑖 ,𝛿
𝑙𝑛
𝑙𝑛
𝑙𝑛
¯ 𝑖,𝛿 𝛿 and 𝑟𝑛𝑙𝑛𝑖 ,𝛿 ∈ 𝑃𝑛 ; moreover, 𝜆0,𝛿 > 0
combination 𝜆𝑛𝑛 𝑝𝑖,𝛿
=𝜆
𝑛 + (1 − 𝜆𝑛 )𝑟𝑛 , with 𝜆𝑛
𝑛
𝑛
0,𝛿
0
¯
since by definition 𝑙𝑛 is the first level 𝑙 where a strategy in 𝑇𝑛 (𝛿) appears in in the support
𝑙1 ,𝛿
of the 𝜎
¯𝑛𝑙,𝛿 . Also, since 𝑙𝑛1,𝛿 is the first level of Λ𝛿 that does not have its support in 𝑃𝑛0 , 𝜋
¯𝑛1 (𝑞𝑛𝑛 )
equals 𝜋𝑛1,𝛿 .
0,𝛿
We claim that for each 𝑙 ⩽ 𝑙𝑛0,𝛿 , 𝑞𝑛𝑙,𝛿 induces the equilibrium outcome against 𝑞𝑚
. Indeed,
0
0,𝛿
∗
0
0,𝛿
since 𝑞𝑚 belongs to 𝑄 , the strategies in 𝑆𝑛 ∪ 𝑇𝑛 (𝛿) are optimal against 𝑞𝑚 ; also, by quasiperfection, every strategy 𝑠1𝑛 in 𝑆𝑛1 , (resp. every strategy 𝑡1𝑛 (𝛿) in 𝑇𝑛1 (𝛿)) such that 𝑠1𝑛 (resp.
0,𝛿
, since 𝑛
𝑡1𝑛 (𝛿, 𝑤𝑛1 ) for some 𝑤𝑛1 ) appears at a level 𝑙 ⩽ 𝑙𝑛0,𝛿 of Λ̄𝛿𝑛 must be a best reply to 𝑞𝑚
0
1
1
has to reject the strategies in 𝑇𝑛 (𝛿) before choosing a strategy in 𝑆𝑛 ∪ 𝑇𝑛 (𝛿). Thus for all
0,𝛿
0,𝛿
0,𝛿
𝑙 ⩽ 𝑙𝑛0,𝛿 , 𝑞𝑛𝑙,𝛿 is a best reply to 𝑞𝑚
. Since 𝑞𝑚
is a lexicographic best reply to Λ𝛿𝑛 , (˜
𝑞𝑛 (𝜀), 𝑞𝑚
)
is an equilibrium of the game Γ for all small 𝜀, where
0,𝛿
0,𝛿
𝑞˜𝑛 (𝜀) = (1 − 𝜀) 𝑞𝑛0,𝛿 + 𝜀((1 − 𝜀)𝑞𝑛1,𝛿 + 𝜀2 ((1 − 𝜀)𝑞𝑛2,𝛿 + 𝜀3 (⋅ ⋅ ⋅ + 𝜀𝑙𝑛 𝑞𝑛𝑙𝑛 ))) .
By genericity of payoffs, Γ has finitely many equilibrium outcomes, so each of these equilibria
induces the same equilibrium outcome—hence the claim follows. Three implications of this
𝑙0 ,𝛿
claim are: (i) 𝑙𝑛1,𝛿 > 𝑙𝑛0,𝛿 ; (ii) the enabling strategy 𝑟𝑛𝑛 in the previous paragraph belongs
to 𝑃𝑛0 ; (iii) all levels up to 𝑙𝑛0,𝛿 prescribe the same mixture at each information set on the
equilibrium path and differ only at information sets excluded by (all of) 𝑚’s equilibrium
strategies in 𝑄∗ .
5.11. Limit of the Lexicographic Probability System. Next we characterize the limit
of the LPS as 𝛿 ↓ 0.
Take a subsequence of the 𝛿’s such that the following properties hold for the associated
LPSs Λ𝛿𝑛 for each 𝑛: (i) 𝑙𝑛𝑖,𝛿 is independent of 𝛿 for each 𝑖, call it 𝑙𝑛𝑖 ; (ii) for each 𝑙 ⩽ 𝑙𝑛1 , the
face of 𝑃𝑛 that contains 𝑞𝑛𝑙,𝛿 in its interior, as well as the strategies for 𝑚 in 𝑆𝑚 that are best
replies to 𝑞𝑛𝑙,𝛿 , are independent of 𝛿.
𝑙1 ,𝛿
𝑙1 ,𝛿
Let 𝜎𝑛𝑛 be a sequence of strategies in Σ𝑛 that is equivalent to the sequence 𝑞𝑛𝑛 . Again
using Blume, Brandenburger, and Dekel [1, Appendix Proposition 2], there is now an LPS
𝑙1 ,0
𝑙1 ,𝑘𝑛
(𝜎𝑛𝑛 , . . . , 𝜎𝑛𝑛
) and a sequence (𝜇0 (𝛿), . . . , 𝜇𝑘𝑛 −1 (𝛿)) ∈ ℝ𝑘𝑛 converging to zero such that for
𝑙1 ,𝛿
a subsequence of 𝛿’s, 𝜎𝑛𝑛 is expressible as the nested combination
1
1
1
1
1
𝜎𝑛𝑙𝑛 ,𝛿 = (1 − 𝜇0 (𝛿))(𝜎𝑛𝑙𝑛 ,0 + 𝜇0 (𝛿)((1 − 𝜇1 (𝛿))𝜎𝑛𝑙𝑛 ,1 + 𝜇1 (𝛿)((1 − 𝜇2 (𝛿))𝜎𝑛𝑙𝑛 ,2 + ⋅ ⋅ ⋅ + 𝜇𝑘𝑛 −1 (𝛿)𝜎𝑛𝑙𝑛 ,𝑘𝑛 ))
𝑙1 ,0
𝑙1 ,𝑘𝑛
This LPS induces an equivalent LPS (𝑞𝑛𝑛 , . . . , 𝑞𝑛𝑛
) in enabling strategies.
AXIOMATIC EQUILIBRIUM SELECTION
21
𝑙1 ,𝑘
Let 𝑘𝑛1 be the first level 𝑘 of this LPS where 𝑞𝑛𝑛 does not belong to 𝑃𝑛0 . Recall from the
𝑙1 ,𝛿
previous section that 𝑞𝑛𝑛
𝑙1 ,𝛿
𝑙1 ,𝛿
𝑙1 ,𝛿
𝑛
𝑛
is expressible as a convex combination 𝜆𝑛𝑛 𝑝1,𝛿
𝑛 + (1 − 𝜆𝑛 )𝑟𝑛
𝑙1 ,𝛿
𝑙1 ,𝛿
and that 𝜋
¯𝑛1 (𝑞𝑛𝑛 ) = 𝜋𝑛1,𝛿 . Express 𝑟𝑛𝑛 as a convex combination 𝛼𝑛0,𝛿 𝑟𝑛0,𝛿 + 𝛼𝑛1,𝛿 𝑟𝑛1,𝛿 , where for
𝑙1 ,𝛿
𝑙1 ,𝛿
𝑙1 ,𝛿
𝑖 = 0, 1, 𝑟𝑛𝑖,𝛿 ∈ 𝑃𝑛𝑖 . Then 𝜆𝑛𝑛 + (1 − 𝜆𝑛𝑛 )𝛼𝑛1,𝛿 > 0 since 𝑞𝑛𝑛 does not belong to 𝑃𝑛0 . Going to
𝑙1 ,𝛿
−1 𝑙1 ,𝛿
𝑙1 ,𝛿
a subsequence, let 𝜆1𝑛 be the limit of (𝜆𝑛𝑛 + (1 − 𝜆𝑛𝑛 )𝛼𝑛1,𝛿 ) 𝜆𝑛𝑛 and let 𝑟𝑛1,0 be the limit of
𝑙1 ,𝑘1
1,0
𝑛 𝑛
𝑟𝑛1,𝛿 . Since the limit of 𝑝1,𝛿
is expressible as a convex combination
𝑛 is 𝑝𝑛 , we have that 𝑞𝑛
1 1,0
1 1,0
0
𝜁𝑛 (𝜆𝑛 𝑝𝑛 + (1 − 𝜆𝑛 )𝑟𝑛 ) + (1 − 𝜁𝑛 )ˇ
𝑝𝑛 for some 𝜁𝑛 > 0 and 𝑝ˇ0𝑛 ∈ 𝑃𝑛0 . Moreover, since 𝜋𝑛1,0 is
1
𝑙1 ,𝑘𝑛
the limit of 𝜋𝑛1,𝛿 , 𝜋
¯𝑛1 (𝑞𝑛𝑛
) = 𝜋𝑛1,0 .
1 +1,𝛿
𝑙1 +𝑘𝑛
𝑞𝑛0,𝛿 , . . . , 𝑞ˆ𝑛𝑛
For each 𝛿 in the subsequence used above, define now an LPS Λ̂𝛿𝑛 = (ˆ
follows: 𝑞ˆ𝑛0,𝛿 = 𝑞𝑛0,0 , 𝑞ˆ𝑛𝑙,𝛿 = 𝑞𝑛𝑙−1,𝛿 if 0 < 𝑙 ⩽ 𝑙𝑛1 , and 𝑞ˆ𝑛𝑙,𝛿 =
𝑙1 ,𝑙−𝑙1 −1
𝑞𝑛𝑛 𝑛
) as
otherwise. The strategy 𝑞ˆ𝑛𝑙,𝛿
𝑙0 +1,𝛿
is independent of 𝛿 for 𝑙 = 0 and 𝑙 > 𝑙𝑛1 . Each level 𝑙 < 𝑙𝑛1 + 𝑘𝑛1 + 1 is a strategy in 𝑃𝑛0 . 𝑞ˆ𝑛𝑛
𝑙0 ,𝛿
𝑙0 ,𝛿
𝑙0 ,𝛿
𝑛
𝑛
is a convex combination 𝜆𝑛𝑛 𝑝0,𝛿
𝑛 + (1 − 𝜆𝑛 )𝑟𝑛
1 +1,𝛿
𝑙1 +𝑘𝑛
while 𝑞ˆ𝑛𝑛
is a convex combination
𝑙1 +𝑘1 ,𝛿
𝜋
¯𝑛1 (ˆ
𝑞𝑛𝑛 𝑛 )
1 1,0
𝜁𝑛 (𝜆1𝑛 𝑝1,0
𝑝0𝑛 where 𝜁𝑛 > 0 and
= 𝜋𝑛1,0 . The next lemma
𝑛 + (1 − 𝜆𝑛 )𝑟𝑛 ) + (1 − 𝜁𝑛 )ˇ
sets out the key properties of Λ̂𝛿𝑛 that lead to a conclusion of our proof.
Lemma 5.4. For all small 𝛿, the LPS Λ̂𝛿𝑛 satisfies the following properties.
(1) A strategy is at least as good a reply against Λ𝛿𝑛 as another only if it is at least as
good a reply against Λ̂𝛿𝑛 .
0,0
(2) If 𝜆1𝑛 < 1, then the strategy 𝑟𝑛1,0 is a best reply to 𝑞𝑚
and is at least as good a reply
lexicographically against Λ̂𝛿𝑚 as every strategy in 𝑆𝑛1 .
0,0
.
(3) 𝜆1𝑛 > 0 and every level 𝑙 < 𝑙𝑛1 + 𝑘𝑛1 + 1 induces the equilibrium outcome against 𝑞𝑚
𝑙0 ,𝛿
𝑙0 ,𝛿
(4) The strategy 𝑞ˆ𝑛𝑙,𝛿 for 𝑙 < 𝑙𝑛0 and the strategy 𝑟𝑛𝑛 if 𝜆𝑛𝑛 < 1 are lexicographic best
𝛿
replies against 𝑞ˆ𝑚
.
Proof of Lemma. Suppose 𝑠𝑚 is a better reply against Λ̂𝛿𝑛 than another strategy 𝑡𝑚 . We
show that 𝑠𝑚 is also a better reply against Λ𝛿𝑛 . Let 𝑙 be the first level of Λ̂𝛿𝑛 such that 𝑠𝑚 is
𝑙,𝛿
a better reply against 𝑞ˆ𝑚
than 𝑡𝑚 . If 𝑙 = 0 then for all small 𝛿, 𝑠𝑚 is a better reply against
0,𝛿
0,𝛿
𝑞𝑛 since 𝑞ˆ𝑛 equals the limit 𝑞𝑛0,0 of 𝑞𝑛0,𝛿 ; thus against Λ𝛿𝑛 , 𝑡𝑚 is a worse reply against the
very first level. If 0 < 𝑙 < 𝑙𝑛1 + 1 then obviously 𝑠𝑚 is better reply against Λ𝛿𝑛 than 𝑡𝑚 since
level 𝑙 of Λ̄𝛿𝑛 corresponds to level 𝑙 − 1 of Λ𝛿𝑛 . Suppose then that 𝑙𝑛1 + 1 ⩽ 𝑙 ⩽ 𝑙𝑛1 + 𝑘𝑛1 + 1.
𝑙1 ,𝛿
𝑙1 ,𝛿
Then for all small 𝛿, 𝑠𝑚 is a better reply against 𝑞𝑛𝑛 since 𝑞𝑛𝑛 is a nested combination of
𝑙1 ,0
𝑙1 ,𝑘
(𝑞𝑛𝑛 , . . . , 𝑞𝑛𝑛 𝑛 ). Thus, 𝑠𝑚 is a better reply against Λ𝛿𝑛 . This proves (1).
In the game Γ̃𝛿 player 𝑛, when finally making a choice among the strategies in 𝑆𝑛1 ∪ 𝑇𝑛1 (𝛿),
would choose a strategy 𝑠𝑛 in 𝑆𝑛1 with positive probability only if it is at least as good a
reply against Λ𝛿𝑛 as the other strategies in 𝑆𝑛1 . Therefore, such a strategy would show up
with positive probability under level 𝑙𝑛1 of Λ𝛿𝑛 (and hence in Λ̂𝛿𝑛 ) only if this is the case. This
22
SRIHARI GOVINDAN AND ROBERT WILSON
𝑙1 ,𝛿
implies that if (1 − 𝜇𝑛𝑛 )𝛼𝑛1 is positive, then the strategy 𝑟𝑛1,𝛿 is at least as good a reply
against Λ𝛿𝑛 as the strategies in 𝑆𝑛1 . Recall that 𝑟𝑛1,0 is the limit of 𝑟𝑛1,𝛿 and 𝜆1𝑛 is the limit
𝑙1 ,𝛿
−1 𝑙1 ,𝛿
1
of (𝜇𝑛𝑛 + (1 − 𝜇𝑙𝑛 ,𝛿 )𝛼𝑛1,𝛿 ) 𝜇𝑛𝑛 . Therefore, by point (1) of this lemma, 𝑟𝑛1,0 is at least as
good a reply against Λ̂𝛿𝑛 as strategies in 𝑆𝑛1 if 𝜆1𝑛 < 1. To show that it is also a best reply
0,0
against 𝑞𝑚
, suppose to the contrary that it is not. Then every strategy in 𝑇𝑛1 (𝛿), regardless
0,𝛿
of outsider 𝑜1𝑛,1 ’s choice, is a better reply against 𝑞𝑚
than 𝑟𝑛1,𝛿 for all small 𝛿, since for 𝛿 = 0
0,0
the strategies in 𝑇𝑛1 (𝛿) are in 𝑃𝑛0 , which are best replies to 𝑞𝑚
. Therefore, quasi-perfection
1
implies that 𝑛 would prefer to play the strategies in 𝑇𝑛 (𝛿) rather than implementing 𝑟𝑛1,0 (or
1
𝑟𝑛1,𝛿 when 𝛿 is small), which shows that (1 − 𝜇𝑙𝑛 ,𝛿 )𝛼𝑛𝑙,𝛿 = 0 for all small 𝛿 and hence that
𝜆1𝑛 = 1. This proves (2).
We turn now to (3). Every strategy 𝑞ˆ𝑛𝑙,𝛿 for 𝑙 < 𝑙𝑛1 +𝑘𝑛1 +1 belongs to 𝑃𝑛0 and is thus optimal
0,0
against 𝑞𝑚
, which belongs to 𝑄∗𝑚 . The strategy 𝑝ˇ0𝑛 (which recall is part of the expression
𝑙1 ,𝑘1
defining 𝑞𝑛𝑛 𝑛 ) which is chosen with positive probability is also in 𝑃𝑛0 and hence optimal
0,0
against 𝑞𝑚
. As we saw in the previous paragraph, if 𝜆1𝑛 < 1, the strategy 𝑟𝑛1 must also be
0,0
0,0
0,𝛿
optimal against 𝑞𝑚
. Obviously 𝑞𝑚
is optimal against Λ̂𝛿𝑛 since it is the limit of 𝑞𝑚
, which
𝛿
0,0
by point (1) is optimal against Λ̂𝑛 . Therefore, for all small 𝜀, (𝑞𝑚 , 𝑞𝑛 (𝜀)) is an equilibrium
of Γ, where
𝑞˜𝑛 (𝜀) =
1
( 𝑙𝑛1∑
+𝑘𝑛
𝑙
𝜀 + (1 − 1𝜆1𝑛 >0 )𝜀
1 +𝑘 1 +1
𝑙𝑛
𝑛
1
)−1 ( 𝑙𝑛1∑
+𝑘𝑛
)
𝜀𝑙 𝑞ˆ𝑛𝑙,𝛿
+ (1 −
1
1
1
1
1𝜆1𝑛 >0 )𝜀𝑙𝑛 +𝑘𝑛 +1 𝑞ˆ𝑛𝑙𝑛 +𝑘𝑛 +1
𝑙=0
𝑙=0
𝑙1 +𝑘1 +1
where 1𝜆1𝑛 >0 is the indicator function. This is impossible if 𝜆1𝑛 = 0, since 𝑞ˆ𝑛𝑛 𝑛 is a
convex combination of a strategy in 𝑃𝑛0 and one in 𝑃𝑛1 . Thus 𝜆1𝑛 > 0. Moreover, since this
is a continuum of equilibria, genericity implies that all of them induce the same outcome.
Therefore, all strategies at levels preceding 𝑙𝑛1 +𝑘𝑛1 +1 induce the equilibrium outcome against
0,0
𝑞𝑚
.
Lastly we prove (4). By the previous paragraph, all strategies in 𝑃𝑛0 are optimal against
𝑘,𝛿
1
1
𝑞ˆ𝑚
for 𝑘 ⩽ 𝑙𝑚
+ 𝑘𝑚
. Thus the optimality of a strategy in 𝑃𝑛0 depends on how it fares against
1 +1
𝑙1 +𝑘𝑚
𝑞ˆ𝑚𝑚
. Obviously every strategy 𝑡𝑛 in the support of 𝑞𝑛0,0 is optimal against Λ𝛿𝑚 for all
1 +1
𝑙1 +𝑘𝑚
small 𝛿 and is therefore optimal against 𝑞ˆ𝑚𝑚
1 +1
𝑙1 +𝑘𝑚
against 𝑞ˆ𝑚𝑚
. Now, if a strategy 𝑠𝑛 ∈ 𝑆𝑛0 is not optimal
, then for all small 𝛿, the strategy 𝑡0𝑛 (𝛿) for some 𝑡0𝑛 in the support of 𝑞𝑛0,0
𝑙1 +𝑘1 +1
is a superior reply to 𝑞ˆ𝑚𝑚 𝑚 regardless of what outsider 𝑜0𝑛,1 ’s choice is. Therefore, in Γ̃𝛿 ,
when player 𝑛, after making a provisional choice of 𝑠𝑛 , reconsiders his decision, he would
prefer to play 𝑡0𝑛 (𝛿) rather than 𝑠𝑛 ; moreover, at every information set where he is choosing
among the strategies in 𝑇𝑛0 (𝛿), he would prefer to play 𝑡0𝑛 (𝛿) rather than the duplicate 𝑠0𝑛 (𝛿)
AXIOMATIC EQUILIBRIUM SELECTION
23
involving 𝑠𝑛 . Hence, the probability of 𝑠𝑛 is zero for all 𝑙 < 𝑙𝑛0 in Λ𝛿𝑛 and its probability under
𝑙0 ,𝛿
𝑟𝑛𝑛 is zero as well. Point (4) now follows.
□
5.12. Final Step of the Proof. The proof can now be completed by invoking the results
established above.
Fix a small 𝛿 that satisfies the properties enumerated in Lemma 5.4 of the previous subsection. Let
)
( ∑ )−1 ( ∑
𝑙 𝑙,𝛿
𝑙
0
𝜀 𝑞ˆ𝑛
𝜀
𝑞¯𝑛 (𝜀) =
0 +1
𝑙⩽𝑙𝑛
0 +1
𝑙⩽𝑙𝑛
1
1
)−1 ( 𝑙𝑛1 +𝑘
)
( 𝑙𝑛1 +𝑘
𝑛 +1
𝑛 +1
∑
∑
1
𝑙
𝑙 𝑙,𝛿
𝑞¯𝑛 (𝜀) =
𝜀
𝜀 𝑞ˆ𝑛 .
0 +2
𝑙=𝑙𝑛
0 +2
𝑙=𝑙𝑛
Observe that 𝑞¯𝑛0 (𝜀) belongs to 𝑃𝑛0 for all 𝜀 and it is a convex combination of 𝑝0,𝛿
𝑛 and a
0
𝛿
subset 𝑅𝑛 of strategies that are at least as good replies against Λ̂𝑚 as other strategies in 𝑆𝑛0 .
1
0
Likewise 𝑞¯𝑛1 (𝜀) is a convex combination of 𝑝1,0
𝑛 , 𝑟𝑛 and a point in 𝑃𝑛 such that the strategy
𝑟𝑛1 if it has a positive weight is at least as good a reply against Λ̂𝛿𝑚 as other strategies in 𝑃𝑛1 .
There exists a small 𝜀 such that a strategy 𝑠𝑚 is at least as good as a strategy 𝑡𝑚 against Λ̂𝛿𝑛
iff it is lexicographically at least as good a reply against the LPS (𝑞 0,0 , 𝑞¯𝑛0 (𝜀), 𝑞¯𝑛1 (𝜀)). Since
𝜋
¯𝑛1 (¯
𝑞𝑛1 (𝜀)) = 𝜋
¯𝑛1 (𝑞𝑛1,0 ) = 𝜋 1,0 for all 𝜀, it follows that (𝑞 0,0 , (𝑝0,𝛿 , 𝑝1,0 ), 𝜋 1,0 ) belongs to 𝒬. As
argued in subsections 5.2 and 5.3, proving that this point belongs to 𝒬 shows that in fact it
belongs to 𝑉 (𝑥∗ ) and hence that 𝑞 0,0 in 𝑈 (𝑞 0,∗ ).
This completes the proof of Theorem 5.1 when 𝑆𝑛1 is not empty for either player 𝑛. In case
𝑆𝑛1 is empty for exactly one player 𝑛, as we said initially in the description of ℙ and 𝒬, we do
not have the factor 𝑃𝑛1 or Π1𝑛 . In the family of games Γ(𝛿, 𝑝ˆ), player 𝑛 decides provisionally
in the first stage on the strategy in 𝑆𝑛 to play and in the second stage gets to execute it or
switch to playing a strategy in 𝑇𝑛 (𝛿, 𝑝ˆ). In the metagame, we do not have outsiders 𝑜1𝑛,𝑗 for
𝑗 = 1, 2, 3. The rest of the proof is essentially the same modulo these provisions.
6. Concluding Remarks
Like our article [13] on forward induction, the characterization in this paper is a step toward
a theory of equilibrium refinement using axioms adapted from decision theory. Theorem 5.1
is confined to games in extensive form with perfect recall, two players, and generic payoffs,
but it suggests that an extension to more general games might be possible.
Previously, some proposed refinements selected equilibria with one or more desirable properties, like admissibility, subgame perfection, or sequential rationality. Other proposed refinements derived some properties from limits of equilibria of games with perturbed strategies, such as perfect, quasi-perfect, and proper equilibria. However, a key step forward was
24
SRIHARI GOVINDAN AND ROBERT WILSON
Kohlberg and Mertens’ [17] argument that an axiomatic development requires set-valued
refinements. Their program achieved remarkable success with Mertens’ [22] definition of a
stable set, which has the desirable properties 1, 2, 3 listed in Section 1 and others too, such
as ordinality [25] and immunity to splitting players into agents.
However, an axiomatic theory of refinement should be based on basic principles of rational
behavior in the game at hand, as in decision theory. This precludes reliance on perturbed
games obtained by perturbing players’ payoffs or strategies. The challenge, therefore, has
been to establish why consideration of perturbed games yields the requisite decision-theoretic
properties.
Our answer here begins with Axiom C, which generalizes the invariance criterion of
Kohlberg and Mertens’ [17] and the small worlds criterion of Mertens’ [24], as explained
in subsection 3.3. Absent a strong invariance property like Axiom C, a refinement is vulnerable to ‘framing effects’ depending on wider contexts in which the given game might be
embedded. In decision theory, such effects were examined by Savage [33], and in cognitive
psychology they play a prominent role in interpreting decisions by subjects in experiments,
as for instance in Kahneman and Tversky [16]. For a theory of thoroughly rational behavior,
however, an axiom should exclude framing effects. Axiom C does this by requiring a solution
of a game to be consistent with the solution of any metagame in which it is embedded. As
shown in Proposition 3.4, it is already true of any equilibrium that it is consistent with an
equilibrium of any metagame in which the game is embedded. Axiom C merely extends to
refinements this fundamental invariance property of equilibria.
Our answer continues with the proof of Theorem 5.2 in Appendix B. There it is shown
that a set 𝑄∗ of equilibria in enabling strategies is stable iff the corresponding projection
map from the pseudomanifold 𝒬 to the space ℙ of enabling strategies is essential. Using this
key property, the proof of Theorem 5.1 shows that for each equilibrium in a component of
undominated equilibria there exists a corresponding metagame for which the equilibrium is
the image of a quasi-perfect equilibrium in the metagame if and only if the projection map
is essential. Hence Axioms A, B, C imply that a solution is a stable set, and conversely due
to Mertens’ previous proofs.
The answer to the ‘why’ question above is thus that, given Axioms A and B, stability with
respect to perturbed games is equivalent to an analogous ‘stability’ with respect to embedding
in metagames, as required by Axiom C. Because in practice every game is embedded in some
wider context, we view Axiom C’s requirement that a refinement is immune to presentation
effects as the relevant criterion from the perspective of decision theory. This view is reinforced
by the facts that Nash equilibria satisfy Axiom C, and that together with Axioms A and B,
the implied refinement agrees with stability based on perturbed games.
AXIOMATIC EQUILIBRIUM SELECTION
25
For a refinement satisfying the axioms, Theorem 5.1 establishes that a solution of a game
must be a component of its undominated equilibria, and that the component must be essential. Because payoffs are assumed to be generic, all equilibria in the component have
the same paths of equilibrium play and thus the same distribution of outcomes. Therefore
the main implication for equilibrium refinements is that a predicted outcome distribution
should result from equilibria in an essential component of the undominated equilibria, and
in particular, from the quasi-perfect and sequential equilibria it necessarily contains. A secondary implication is that after deviations from equilibrium play, the continuations of all
equilibria in the component remain admissible and sequentially rational, where those that
are not sequential equilibria of the original game are justified by beliefs induced by quasiperfect equilibria of corresponding metagames that embed the given game. This resolves the
conundrum posed by Reny [30, 31, 32].
Appendix A. Enabling Strategies
In the normal-form representation of a game in extensive form, a player’s pure strategy
specifies the actions chosen at his information sets in the game tree. However, outcomes
are not affected by a strategy’s actions at information sets excluded by his previous actions.
One therefore considers equivalence classes of pure strategies. Say that two pure strategies
are outcome equivalent if the sets of terminal nodes they do not exclude are the same. For
instance, the game in Figure 1 is shown on the left side in extensive form and on the right
side in the ‘pure reduced normal form’ (PRNF) introduced by Mailath, Samuelson, and
Swinkels [21]. In the PRNF each outcome-equivalent class of player 1’s pure strategies is
identified by the terminal nodes it does not exclude, as indicated by labels of rows along the
left side; and each equivalence class of player 2’s pure strategies is identified by the terminal
nodes it does not exclude, as indicated by labels of columns along the top. Because this
game has no moves by Nature, each row and column determine a unique outcome that is the
intersection of the row and column labels, shown as the corresponding entry in the matrix.
A similar example is shown in Figure 2 for the game tree of a signaling game. In this case,
each profile of pure strategies determines a pair of outcomes such that the first or second
outcome occurs depending on whether Nature’s initial move is up or down. For instance,
the outcome of 1’s strategy 𝑎𝑏𝑐𝑑 and 2’s strategy 𝑎𝑐𝑒𝑔 is 𝑎 with probability 𝑝 and 𝑐 with
probability 1 − 𝑝.
Say that a terminal node that is not excluded is an enabled outcome. A pure strategy of
a player enables outcome 𝑧 if it chooses all his actions on the path to 𝑧. A player’s mixed
strategy randomizes over his pure strategies, whereas a behavioral strategy randomizes over
actions at each of his information sets. A strategy of either kind induces a probability
distribution over outcome-equivalent classes of his pure strategies, and thus a distribution
over enabled outcomes. Such a distribution is called an enabling strategy. A point 𝑝𝑛 ∈ [0, 1]𝑍
26
SRIHARI GOVINDAN AND ROBERT WILSON
1
2
1
a
b
1
2
c
e
1 — 2:
ac
aef
bd
bgh
ab
a
a
b
b
cdeg
c
e
d
g
cdfh
c
f
d
h
d
g
f
h
Figure 1. A game tree and its pure reduced normal form in which each pure
strategy is identified by the terminal nodes it does not exclude.
a
e
1
b
f
p
2
2
1−p
c
d
1
g
1—2:
abcd
abgh
efcd
efgh
aceg acfh bdeg bdfh
ac
ac
bd
bd
ag
ah
bg
bh
ec
fc
ed
fd
eg
fh
eg
fh
h
Figure 2. The game tree of a signaling game and its pure reduced normal
form in which each pure strategy is identified by the terminal nodes it does
not exclude.
is an enabling strategy for player 𝑛 if it is the distribution over enabled outcomes induced
by some mixed or behavioral strategy, i.e. 𝑝𝑛 (𝑧) is the mixed strategy’s probability of those
pure strategies that enable outcome 𝑧. The vertices of the polyhedron 𝑃𝑛 of 𝑛’s enabling
strategies correspond to outcome-equivalent classes of 𝑛’s pure strategies in the PRNF, as
in Figures 1 and 2. Enabling strategies are minimal representations of strategic behavior in
games with perfect recall.6
Let 𝑝∗ (𝑧) be the probability that Nature’s strategy enables outcome 𝑧, which is 1 if Nature
∏
has no moves. Then for each profile 𝑝 ∈ 𝑃 = 𝑛 𝑃𝑛 of players’ enabling strategies, the
∏
probability that outcome 𝑧 results is 𝛾𝑧 (𝑝) = 𝑝∗ (𝑧) 𝑛 𝑝𝑛 (𝑧), because Nature and the players
randomize independently. The extensive form is therefore summarized by the multilinear
function 𝛾 : 𝑃 → Δ(𝑍) ⊂ ℝ𝑍 that assigns to each profile of players’ enabling strategies a
distribution over terminal nodes, including the effect of Nature’s enabling strategy. Player
∑
𝑛’s expected payoff is 𝒢𝑛 (𝑝) = 𝑧 𝛾𝑧 (𝑝)𝑢𝑛 (𝑧). The game Γ is therefore summarized by the
6Mertens
[24, p. 554] introduces the mapping of mixed strategies to induced distributions on terminal
nodes. Koller and Megiddo [18] call them realization plans.
AXIOMATIC EQUILIBRIUM SELECTION
27
multilinear function 𝒢 : 𝑃 → ℝ𝑁 that assigns to each profile of players’ enabling strategies
their expected payoffs. This summary specification is called the enabling form of the game.
Appendix B. Proofs of Propositions
B.1. Proof of Proposition 3.3. A multilinear map 𝑓𝑛 : Σ̃𝑛 × Σ𝑜 → Σ𝑛 is completely
specified by its values at profiles of pure strategies. We use 𝑓ˆ𝑛 to denote the restriction of
𝑓𝑛 to the set 𝑆˜𝑛 × 𝑆𝑜 of profiles of pure strategies.
˜ embeds 𝐺 via a collection of multilinear maps 𝑓 = (𝑓𝑛 )
Proposition B.1. 𝐺
𝑛∈𝑁 if and only
if for each player 𝑛 there exists 𝑇˜𝑛 ⊆ 𝑆˜𝑛 and a bijection 𝜋𝑛 : 𝑇˜𝑛 → 𝑆𝑛 such that for each
(˜
𝑠, 𝑠𝑜 ) ∈ 𝑆˜ × 𝑆𝑜 and 𝑡˜𝑛 ∈ 𝑇˜𝑛 :
(1)
(2)
𝑓ˆ𝑛 (𝑡˜𝑛 , 𝑠𝑜 ) = 𝜋𝑛 (𝑡˜𝑛 ),
˜ 𝑛 (˜
𝐺
𝑠, 𝑠𝑜 ) = 𝐺𝑛 (𝑓ˆ(˜
𝑠, 𝑠𝑜 )), where 𝑓ˆ = (𝑓ˆ𝑛 )𝑛∈𝑁 .
˜ : Σ̃ × Σ𝑜 → ℝ𝑁 ∪ 𝑜 and a collection of multilinear maps
Proof. Suppose we have a game 𝐺
𝑓𝑛 : Σ̃𝑛 × Σ𝑜 → Σ𝑛 , one for each 𝑛 ∈ 𝑁 , such that conditions (1) and (2) of the proposition
are satisfied. Then, by condition (1) and multilinearity of 𝑓𝑛 for each 𝑛, for each fixed
𝜎𝑜 , 𝑓𝑛 (⋅, 𝜎𝑜 ) is surjective because it maps the face spanned by 𝑇˜𝑛 homeomorphically onto
˜ = 𝐺 ∘ 𝑓 . According to
Σ𝑛 . Also, condition (2) and multilinearity of each 𝑓𝑛 imply that 𝐺
˜ 𝑓 ) embeds 𝐺.
Definition 3.2, therefore, (𝐺,
˜ 𝑓 ) embeds 𝐺. Let 𝜎𝑜 be a profile of completely mixed strategies for
Now suppose that (𝐺,
outsiders. Because 𝑓𝑛 is multilinear it induces a linear mapping 𝑓𝑛 (⋅, 𝜎𝑜 ) from Σ̃𝑛 to Σ𝑛 that,
by the definition of an embedding, is surjective. Hence, for each 𝑠𝑛 ∈ 𝑆𝑛 there exists a pure
strategy 𝑡˜𝑛 (𝑠𝑛 ) in 𝑆˜𝑛 that is mapped to 𝑠𝑛 by this linear map. We claim that 𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝑠𝑜 ) =
∑
𝑠𝑛 for all 𝑠𝑜 ∈ 𝑆𝑜 . Indeed, observe that 𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝜎𝑜 ) = 𝑠𝑜 𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝑠𝑜 )𝜎𝑜 (𝑠𝑜 ), where for
each 𝑠𝑜 , 𝜎𝑜 (𝑠𝑜 ) is the probability of 𝑠𝑜 under 𝜎𝑜 . Therefore, since 𝜎𝑜 is completely mixed, if
𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝑠𝑜 ) ∕= 𝑠𝑛 for some 𝑠𝑜 then 𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝜎𝑜 ), which is an average of values at vertices of
𝑆𝑜 , cannot be 𝑠𝑛 . Thus, 𝑓𝑛 (𝑡˜𝑛 (𝑠𝑛 ), 𝑠𝑜 ) = 𝑠𝑛 for all 𝑠𝑜 . Let 𝑇˜𝑛 ⊂ 𝑆˜𝑛 be a collection comprising
a different pure strategy 𝑡˜𝑛 (𝑠𝑛 ) for each 𝑠𝑛 ∈ 𝑆𝑛 and let 𝜋𝑛 be the associated bijection. Define
𝑓ˆ𝑛 : 𝑆˜𝑛 × 𝑆𝑜 → Σ𝑛 by 𝑓ˆ𝑛 (˜
𝑠𝑛 , 𝑠𝑜 ) = 𝑓𝑛 (˜
𝑠𝑛 , 𝑠𝑜 ). Then conditions (1) and (2) of the proposition
are satisfied.
□
B.2. Proof of Proposition 3.4.
˜ 𝑓 ) embeds 𝐺 then the equilibria of 𝐺 are the 𝑓 -images of the
Proposition B.2. If (𝐺,
˜
equilibria of 𝐺.
˜ and let 𝜎 = 𝑓 (˜
Proof. Suppose (˜
𝜎 , 𝜎𝑜 ) is an equilibrium of 𝐺
𝜎 , 𝜎𝑜 ). For any insider 𝑛 and his
strategy 𝜏𝑛 ∈ Σ𝑛 there exists 𝜏˜𝑛 ∈ Σ̃𝑛 such that 𝑓𝑛 (˜
𝜏𝑛 , 𝜎𝑜 ) = 𝜏𝑛 because 𝑓𝑛 (⋅, 𝜎𝑜 ) is surjective
28
SRIHARI GOVINDAN AND ROBERT WILSON
by condition (a) of Definition 3.2 an embedding. Using condition (b),
˜ 𝑛 (˜
˜ 𝑛 (˜
𝐺𝑛 (𝜏𝑛 , 𝜎−𝑛 ) = 𝐺𝑛 (𝑓 (˜
𝜏𝑛 , 𝜎
˜−𝑛 , 𝜎𝑜 )) = 𝐺
𝜏𝑛 , 𝜎
˜−𝑛 , 𝜎𝑜 ) ⩽ 𝐺
𝜎 , 𝜎𝑜 ) = 𝐺𝑛 (𝑓 (˜
𝜎 , 𝜎𝑜 )) = 𝐺𝑛 (𝜎) ,
˜ Hence 𝜎 is an equilibrium
where the inequality obtains because (˜
𝜎 , 𝜎𝑜 ) is an equilibrium of 𝐺.
of 𝐺.
Conversely, suppose 𝜎 is an equilibrium of 𝐺. For each 𝑛, let 𝜋𝑛 be the bijection given by
˜ defined by 𝜎
Proposition 3.3. Let 𝜎
˜𝑛 be the strategy for insider 𝑛 in 𝐺
˜𝑛 (𝑡˜𝑛 ) = 𝜎𝑛 (𝜋𝑛 (𝑡˜𝑛 )) for
𝑡˜𝑛 ∈ 𝑇˜𝑛 and 𝜎
˜𝑛 (˜
𝑠𝑛 ) = 0 for 𝑠˜𝑛 ∈
/ 𝑇˜𝑛 . Since 𝑓𝑛 is multilinear, by condition (1) of Proposition
3.3, 𝑓𝑛 (˜
𝜎𝑛 , ⋅) = 𝜎𝑛 and thus 𝑓 (˜
𝜎 , ⋅) = 𝜎. Hence, it suffices to show that there exists a
˜ By fixing the profile
strategy profile 𝜎𝑜 for outsiders such that (˜
𝜎 , 𝜎𝑜 ) is an equilibrium of 𝐺.
of insiders’ strategies to be 𝜎
˜ one induces a game among outsiders. Let 𝜎𝑜 be an equilibrium
˜ observe
of this induced game among outsiders. To see that (˜
𝜎 , 𝜎𝑜 ) is an equilibrium of 𝐺,
that for each pure strategy 𝑠˜𝑛 of an insider 𝑛:
˜ 𝑛 (˜
˜ 𝑛 (˜
𝐺
𝑠𝑛 , 𝜎
˜−𝑛 , 𝜎𝑜 ) = 𝐺𝑛 (𝑓𝑛 (˜
𝑠𝑛 , 𝜎𝑜 ), 𝜎−𝑛 ) ⩽ 𝐺𝑛 (𝜎) = 𝐺𝑛 (𝑓 (˜
𝜎 , 𝜎𝑜 )) = 𝐺
𝜎 , 𝜎𝑜 ) ,
where the first and second equalities use the property 𝑓 (˜
𝜎 , ⋅) = 𝜎 established above, and the
inequality obtains because 𝜎 is an equilibrium of 𝐺.
□
Appendix C. Proof of Theorem 5.2
Theorem C.1. (𝒬, ∂𝒬) is a pseudomanifold of the same dimension as (ℙ, ∂ℙ). Moreover,
𝑄∗ is stable if and only if the projection map Ψ : (𝒬, ∂𝒬) → (ℙ, ∂ℙ) is essential.
Proof. The proof invokes genericity of payoffs by assuming that certain points and polyhedra, identified as they arise during the proof, are in general position. See Appendix E for
elaboration of these genericity requirements.
For any set 𝑋, we write 𝑑(𝑋) for its dimension. For any subset 𝑇𝑛 of 𝑆𝑛 , let 𝑃𝑛 (𝑇𝑛 ) be
the convex hull (in 𝑃𝑛 ) of the strategies in 𝑇𝑛 . For simplicity, we write 𝑑(𝑇𝑛 ) for 𝑑(𝑃𝑛 (𝑇𝑛 )).
See subsection 5.1 for additional notation used below.
For any vertex 𝜋𝑛1 of Π1𝑛 , let 𝑆𝑛1 (𝜋𝑛1 ) be the set of pure strategies that map to 𝜋𝑛1 under 𝜋
¯𝑛1 .
For any face Ψ1𝑛 of Π1𝑛 , let 𝑆𝑛1 (Ψ1𝑛 ) be the set of pure strategies 𝑠1𝑛 such that 𝜋
¯𝑛1 (𝑠1𝑛 ) ∈ Ψ1𝑛 .
Let 𝐻𝑛0 be the set of information sets ℎ𝑛 ∈ 𝐻𝑛 ∖ 𝐻𝑛∗ of player 𝑛 such that at the last
information set ℎ′𝑛 ∈ 𝐻𝑛∗ that precedes ℎ𝑛 , the action there leading to ℎ𝑛 belongs to 𝐴∗𝑛 ,
which is the set of his equilibrium actions. If a subset 𝑇𝑛0 of 𝑆𝑛0 is such that 𝑃𝑛 (𝑇𝑛0 ) contains
¯ ∗ ≡ 𝜌𝑛 (Σ̄𝑛 ), then for each first information set ℎ𝑛 ∈ 𝐻 0 there exists a
an equilibrium in 𝑄
𝑛
𝑛
strategy in 𝑇𝑛0 that enables ℎ𝑛 . Let 𝐻𝑛1 = 𝐻𝑛 ∖ (𝐻𝑛∗ ∪ 𝐻𝑛0 ).
Lemma C.2. Suppose 𝑇𝑛0 is a subset of strategies in 𝑆𝑛0 such that 𝑃𝑛 (𝑇𝑛0 ) contains an
¯ ∗ . If the strategies in 𝑇 0 are at least as good replies against 𝑝0 ∈ 𝑃 0 as
equilibrium 𝑞𝑛∗ in 𝑄
𝑚
𝑚
𝑛
𝑛
other strategies in 𝑆𝑛0 , then all the strategies in 𝑆𝑛0 are equally good replies against 𝑝0𝑚 .
AXIOMATIC EQUILIBRIUM SELECTION
29
∗
Proof. Let Π∗𝑛 be the projection of 𝑃𝑛0 to ℝ𝑍 , where 𝑍 ∗ is the set of terminal nodes reached
0
¯ ∗ . 𝑍 ∗ = 𝑍𝑛0 ∩ 𝑍𝑚
. Consequently, the
with positive probability under the equilibria in 𝑄
0
0
0
payoff to a strategy 𝑠𝑛 against 𝑝𝑚 depends on 𝑠𝑛 only through its projection to Π∗𝑛 . Since
the projection of 𝑞𝑛∗ —which is at least as good a reply as the other strategies in 𝑆𝑛0 against
𝑝0𝑚 —belongs to the relative interior of Π∗𝑛 , the strategies in 𝑆𝑛0 are equally good replies against
𝑝0𝑚 .
□
Lemma C.3. Suppose that 𝑇𝑛0 is a subset of 𝑆𝑛0 such that 𝑃𝑛 (𝑇𝑛0 ) is a face of 𝑃𝑛0 containing
¯ ∗𝑛 . For each 𝑡𝑛 ∈ 𝑆𝑛0 ∖ 𝑇𝑛0 there exists an information set ℎ𝑛 ∈ 𝐻𝑛0
an equilibrium strategy in 𝑄
that 𝑡𝑛 enables and where the action chosen by it is avoided by all 𝑡0𝑛 ∈ 𝑇𝑛0 that enable ℎ𝑛 .
Proof. Since 𝑃𝑛 (𝑇𝑛0 ) is a face of 𝑃𝑛0 , which is itself a face of 𝑃𝑛 , there exists a linear function
𝑓 : ℝ𝑍 → ℝ that is zero on 𝑃𝑛 (𝑇𝑛0 ) and negative everywhere else on 𝑃𝑛 . Fix 𝑝𝑚 in the interior
of 𝑃𝑚 and define a payoff function 𝑢˜𝑛 for 𝑛 by the equation: 𝑝0 (𝑧)𝑝𝑚 (𝑧)˜
𝑢𝑛 (𝑧) = 𝑓 (𝑒𝑧 ) where
𝑍
𝑒𝑧 is the 𝑧-th unit vector in ℝ . Then when 𝑛’s payoff function is 𝑢˜𝑛 , the strategies in 𝑃𝑛 (𝑇𝑛0 )
are the best replies against 𝑝𝑚 . Take 𝑡𝑛 ∈
/ 𝑇𝑛0 . Since it is suboptimal against 𝑝𝑚 , there exists
an information set ℎ𝑛 where the action 𝑎 chosen by 𝑡𝑛 is suboptimal. Then ℎ𝑛 does not
belong to 𝐻𝑛∗ : indeed, since 𝑡𝑛 belongs to 𝑆𝑛0 , 𝑎 is an equilibrium action if ℎ𝑛 ∈ 𝐻𝑛∗ ; and since
𝑃𝑛 (𝑇𝑛0 ) contains an equilibrium strategy, there exists a strategy in 𝑇𝑛0 that enables such an
ℎ𝑛 and chooses 𝑎. Let ℎ′𝑛 be the last information set preceding ℎ𝑛 that belongs to 𝐻𝑛∗ . Then
obviously the action chosen by 𝑡𝑛 at ℎ′𝑛 is an equilibrium action, as 𝑡𝑛 ∈ 𝑆𝑛0 , and thus ℎ𝑛
belongs to 𝐻𝑛0 . Any strategy in 𝑇𝑛0 that enables ℎ𝑛 avoids 𝑎, which proves the lemma.
□
For the next lemma, it is worth recapitulating the exact definition of the set Π1𝑛 . Recall
from subsection 5.1 that we fix a completely mixed enabling strategy 𝑝¯𝑚 for player 𝑚 and
compute for each 𝑝𝑛 the total probability 𝜂(𝑝𝑛 ) of reaching a terminal node in 𝑍𝑛1 under
1
1
(¯
𝑝𝑚 , 𝑝𝑛 ). ℋ𝑛 is a hyperplane in ℝ𝑍𝑛 that separates the projection Ψ𝑍𝑛1 (𝑃𝑛1 ) of 𝑃𝑛1 to ℝ𝑍𝑛
1
from the origin of ℝ𝑍𝑛 and that has (𝑝0 (𝑧)¯
𝑝𝑚 (𝑧))𝑧∈𝑍𝑛1 as its normal and some 𝜀 > 0 as its
constant. The function 𝜋
¯𝑛1 maps each point 𝑝𝑛 ∈ 𝑃𝑛 ∖ 𝑃𝑛0 to 𝜀(𝜂(𝑝𝑛 ))−1 Ψ𝑍𝑛1 (𝑝𝑛 ) ∈ ℋ𝑛 .
¯𝑛1 (𝑠𝑛 ) is a vertex of Π1𝑛 iff there exists a unique
Lemma C.4. For a strategy 𝑠𝑛 of 𝑆𝑛1 , 𝜋
information set ℎ𝑛 ∈ 𝐻𝑛∗ with the property that 𝑠𝑛 enables ℎ𝑛 and chooses a non-equilibrium
action there.
Proof. Let 𝑠𝑛 ∈ 𝑆𝑛1 be a pure strategy satisfying the condition of the lemma. We prove by
¯𝑛1 (𝑠𝑛 ),
contradiction that 𝜋
¯𝑛1 (𝑠𝑛 ) is a vertex of Π1𝑛 . Therefore suppose to the contrary that 𝜋
¯𝑛1 (𝑠𝑛 ) as a unique
which equals 𝜀(𝜂(𝑠𝑛 ))−1 Ψ𝑍𝑛1 (𝑠𝑛 ), is not a vertex of Π1𝑛 . We can express 𝜋
∑
convex combination 𝐽𝑗=1 𝜆𝑗 𝜋𝑛1,𝑗 where 𝐽 > 1 and for each 1 ⩽ 𝑗 ⩽ 𝐽, 𝜋𝑛1,𝑗 is a vertex of
−1
1,𝑗
1 (𝑠
Π1𝑛 . For each 𝑗, since 𝜋𝑛1,𝑗 is a vertex of Π1𝑛 , we can express 𝜋𝑛1,𝑗 as 𝜀(𝜂(𝑠1,𝑗
𝑛 ) for
𝑛 )) Ψ𝑍𝑛
′
𝑗
1,𝑗
1
1,𝑗
some 𝑠𝑛 in 𝑆𝑛 . Since 𝜆 > 0, 𝑠𝑛 cannot choose a non-equilibrium action at any ℎ𝑛 ∕= ℎ𝑛
30
SRIHARI GOVINDAN AND ROBERT WILSON
in 𝐻𝑛∗ that it enables; since it belongs to 𝑆𝑛1,𝑗 it must therefore enable ℎ𝑛 ; and it cannot
choose a different non-equilibrium action from 𝑠𝑛 at ℎ𝑛 . Observe now that the probability
𝜂(𝑠𝑛 ) and 𝜂(𝑠1,𝑗
𝑛 ) for all 𝑗 are equal and exactly the probability that Nature and the strategy
∑
1,𝑗
1,𝑗
𝑝¯𝑚 do not exclude ℎ𝑛 . Therefore, Ψ𝑍𝑛1 (𝑠𝑛 ) = 𝑗 𝜆𝑗 Ψ𝑍𝑛1 (𝑠1,𝑗
𝑛 ). Modify 𝑠𝑛 to a strategy 𝑡𝑛
so that at every information set other than the successors to ℎ𝑛 , 𝑡1,𝑗
𝑛 agrees with 𝑠𝑛 , and
1,𝑗
at the successors to ℎ𝑛 it agrees with 𝑠𝑛 . It is now clear that when viewed as enabling
∑
1,𝑗
strategies, 𝑠𝑛 = 𝑗 𝜆𝑗 𝑡1,𝑗
𝑛 and thus 𝑠𝑛 is a convex combination of the strategies 𝑡𝑛 . But that
is a contradiction since 𝑠𝑛 is a vertex of 𝑃𝑛 and all the strategies 𝑡1,𝑗
𝑛 are different from one
1,𝑗
another and from 𝑠𝑛 as they induce the points 𝜋𝑛 that are different from one another and
from 𝜋
¯𝑛1 (𝑠𝑛 ) in ℋ𝑛 .
To prove the other way around, suppose 𝑠𝑛 is a strategy that, at a collection ℎ𝑘𝑛 for 𝑘 =
1, . . . , 𝐾 of at least two information sets in 𝐻𝑛∗ , chooses a non-equilibrium action. For each 𝑘,
𝑘
1
choose a strategy 𝑠1,𝑘
𝑛 in 𝑆𝑛 that enables ℎ𝑛 , agrees with 𝑠𝑛 there and at all its successors, but
∑
at other ℎ𝑛 ∈ 𝐻𝑛∗ , chooses an equilibrium action. Then Ψ𝑍𝑛1 (𝑠𝑛 ) = 𝑘 Ψ𝑍𝑛1 (𝑠1,𝑘
𝑛 ). Therefore,
1
1
𝜋
¯𝑛 (𝑠𝑛 ) cannot be a vertex of Π𝑛 .
□
For each 𝑇𝑛0 ⊆ 𝑆𝑛0 such that 𝑃𝑛 (𝑇𝑛0 ) is a face of 𝑃𝑛0 and contains an equilibrium strategy
for 𝑛, let 𝑆𝑛1 (𝑇𝑛0 ) be the subset of 𝑆𝑛1 consisting of strategies 𝑠1𝑛 such that there exists a
strategy 𝑡0𝑛 ∈ 𝑇𝑛0 that agrees with 𝑠0𝑛 at all information sets in 𝐻𝑛∗ and 𝐻𝑛0 that 𝑠1𝑛 enables,
except those in 𝐻𝑛∗ where 𝑠1𝑛 chooses a nonequilibrium action. For a face Ψ1𝑛 of Π1𝑛 , let
𝑆𝑛1 (𝑇𝑛0 ; Ψ1𝑛 ) = 𝑆𝑛1 (Ψ1𝑛 ) ∩ 𝑆𝑛1 (𝑇𝑛0 ) and 𝑇𝑛 ≡ 𝑇𝑛0 ∪ 𝑆𝑛1 (𝑇𝑛0 ; Ψ1𝑛 ). For notational simplicity, we refer
to 𝑆𝑛1 (𝑇𝑛0 ; Ψ1𝑛 ) as 𝑇𝑛1 . The following lemma provides an important feature of the set 𝑇𝑛 .
Lemma C.5. The strategies in 𝑇𝑛 are the vertices of a face of 𝑃𝑛 whose dimension is
𝑑(𝑇𝑛 ) ≡ 𝑑(𝑇𝑛0 ) + 𝑑(Ψ1𝑛 ) + 1.
Proof. Let 𝑇ˆ𝑛1 be the set of strategies 𝑡1𝑛 in 𝑇𝑛1 such 𝜋
¯𝑛1 (𝑡1𝑛 ) is a vertex of Ψ1𝑛 . We will first
show that every 𝑡𝑛 ∈ 𝑇𝑛 ∖ (𝑇𝑛0 ∪ 𝑇ˆ𝑛1 ) is affinely dependent on the strategies in 𝑇𝑛0 ∪ 𝑇ˆ𝑛1 . Let
∗
𝑡1𝑛 ∈ 𝑇𝑛 ∖ (𝑇𝑛0 ∪ 𝑇ˆ𝑛1 ). By Lemma C.4, there exist information sets ℎ1𝑛 , . . . ℎ𝐾
𝑛 , 𝐾 > 1, in 𝐻𝑛
such that for each 𝑘, 𝑡1𝑛 chooses a non-equilibrium action 𝑎𝑘 at ℎ𝑘𝑛 , and at each other ℎ𝑛 ∈ 𝐻𝑛∗
it chooses an equilibrium action. Fix 𝑡0𝑛 ∈ 𝑇𝑛0 that agrees with 𝑡1𝑛 everywhere except at the
information sets ℎ𝑘𝑛 , and their successors, for each 𝑘. For each 𝑘 let 𝑡1,𝑘
𝑛 be the strategy that
𝑘
1
agrees with 𝑡𝑛 at ℎ𝑛 and its successors, but everywhere else agrees with 𝑡0𝑛 . Each 𝑡1,𝑘
𝑛 belongs
∑
1
0
to 𝑇ˆ𝑛1 by Lemma C.3 and also, 𝑡1𝑛 = 𝑘 𝑡1,𝑘
𝑛 − (𝐾 − 1)𝑡𝑛 . Thus, 𝑡𝑛 is an affine combination
of the strategies in 𝑇 0 ∪ 𝑇ˆ1 .
𝑛
𝑛
0
1
¯𝑛1 (𝑡1,𝑗
For each 𝑗 = 0, . . . , 𝑑(Ψ1𝑛 ), pick a strategy 𝑡1,𝑗
𝑛 ) is a vertex of
𝑛 ∈ 𝑆𝑛 (𝑇𝑛 ) such that 𝜋
Ψ1𝑛 . Let 𝑇˜𝑛1 be the collection of these strategies. We show that strategies in 𝑇ˆ𝑛1 ∖ 𝑇˜𝑛1 are
now affinely dependent on the strategies in 𝑇𝑛0 ∪ 𝑇˜𝑛1 . Fix 𝑡1𝑛 ∈ 𝑇ˆ𝑛1 ∖ 𝑇˜𝑛1 . By Lemma C.4
AXIOMATIC EQUILIBRIUM SELECTION
31
there exists a unique information set ℎ𝑛 ∈ 𝐻𝑛∗ enabled by 𝑡1𝑛 and where it chooses a non𝐽
˜1
equilibrium action. By construction of 𝑇˜𝑛1 , there exists a subset (𝑡1,𝑗
𝑛 )𝑗=1 of 𝑇𝑛 such that
∑
𝑗
𝜋
¯𝑛1 (𝑡1𝑛 ) is expressible as an affine combination 𝑗 𝜆𝑗 𝜋
¯ (𝑡1,𝑗
𝑛 ) with 𝜆 ∕= 0 for all 𝑗. For each
∗
𝑗, 𝑠1,𝑗
𝑛 enables ℎ𝑛 and chooses 𝑎 at ℎ𝑛 ; at all other information sets in 𝐻𝑛 it chooses an
equilibrium action. Let 𝑡0𝑛 ∈ 𝑇𝑛0 be a strategy that agrees with 𝑡1𝑛 everywhere except at ℎ𝑛
0
1,𝑗
and its successors. For each 𝑗, let 𝑡˜0,𝑗
𝑛 be a strategy in 𝑇𝑛 that agrees with 𝑡𝑛 everywhere
˜0,𝑗
except at ℎ𝑛 and its successors. Modify 𝑡˜0,𝑗
𝑛 to a strategy that agrees with 𝑡𝑛 everywhere
0
except at ℎ𝑛 and its successors, where it agrees with 𝑡0𝑛 . By Lemma C.3, 𝑡0,𝑗
𝑛 belongs to 𝑇𝑛
∑
0,𝑗
for each 𝑗. Now 𝑡1𝑛 = 𝑡0𝑛 + 𝑗 𝜆𝑗 (𝑡1,𝑗
𝑛 − 𝑡𝑛 ) and is affinely dependent on the strategies in
𝑇𝑛0 ∪ 𝑇˜𝑛1 .
It follows now from the above arguments that the affine space 𝐴 spanned by 𝑇𝑛0 ∪ 𝑇˜𝑛1
contains 𝑃𝑛 (𝑇𝑛 ) and that the dimension of 𝑃𝑛 (𝑇𝑛 ) is as stated. To finish the proof of the
lemma, we show that 𝑃𝑛 (𝑇𝑛 ) is a face of 𝑃𝑛 . Let 𝑄𝑛 be the smallest face of 𝑃𝑛 that contains
𝑃𝑛 (𝑇𝑛 ). Suppose 𝑄𝑛 ∕= 𝑃𝑛 (𝑇𝑛 ). There exists a point 𝑝𝑛 in the relative of interior of 𝑃𝑛 (𝑇𝑛 )
and 𝑄𝑛 . Therefore 𝑝𝑛 can be expressed as a convex combination of the vertices of 𝑄𝑛 in two
∑
∑ 𝑗 1,𝑗
0,𝑖
0
1,𝑗
1
different ways: (a) 𝑖 𝜆𝑖 𝑡0,𝑖
𝑛 +
𝑗 𝜆 𝑡𝑛 , where the 𝑡𝑛 ’s are in 𝑇𝑛 and the 𝑡𝑛 ’s are in 𝑇𝑛 ;
∑
∑ 𝑗 1,𝑗 ∑ 𝑘 2,𝑘
2,𝑘
(b) 𝑖 𝜇𝑖 𝑡0,𝑖
𝑛 +
𝑗 𝜇 𝑡𝑛 +
𝑘 𝜇 𝑡𝑛 , where now the 𝑡𝑛 ’s are the vertices of 𝑄𝑛 that are not
0
0
in 𝑇𝑛 . Consider one of the 𝑡2,𝑘
𝑛 ’s. If it belongs to 𝑆𝑛 ∖ 𝑇𝑛 then by Lemma C.3 there is an
information set ℎ𝑛 in 𝐻𝑛0 that is enabled by 𝑡2,𝑘
𝑛 where the action chosen by ℎ𝑛 is avoided all
0
strategies in 𝑇𝑛 that enable it; by the definition, the strategies in 𝑇𝑛1 avoid this action as well.
This implies under the expression in (b) that the nodes following this action are assigned a
1
1
positive probability, but not under (a), which is impossible. If 𝑡2,𝑘
𝑛 belongs to 𝑆𝑛 ∖ 𝑇𝑛 then
it must belong to 𝑆𝑛1 (Ψ1𝑛 ) since otherwise under (b) 𝜋
¯𝑛1 (𝑝𝑛 ) ∈
/ Ψ1𝑛 . Since 𝑠𝑛 ∈
/ 𝑇𝑛1 there exists
2,𝑘
an information set ℎ𝑛 ∈ 𝐻𝑛0 enabled by 𝑡2,𝑘
𝑛 where the continuation strategy of 𝑡𝑛 coincides
with that of some 𝑡𝑛 ∈ 𝑆𝑛0 ∖ 𝑇𝑛0 but not for any 𝑠𝑛 ∈ 𝑇𝑛0 . As in the previous case, this too is
impossible.
□
One corollary of the above result obtains when we take 𝑇𝑛1 to be 𝑆𝑛1 and Ψ1𝑛 to be Π1𝑛 . The
dimension of 𝑃𝑛 is 𝑑(𝑃𝑛0 ) + 𝑑(Π1𝑛 ) + 1. Observe that 𝑃𝑛1 is a face of 𝑃𝑛 iff Ψ𝑍𝑛1 (𝑃𝑛1 ) is a face
of Ψ𝑍𝑛1 (𝑃𝑛 ), which is equivalent to saying that Π1𝑛 is homeomorphic to Ψ𝑍𝑛1 (𝑃𝑛1 ). Thus, the
dimension of 𝑃𝑛1 equals 𝑑(Π1𝑛 ) if 𝑃𝑛1 is a face of 𝑃𝑛 and otherwise it equals the dimension of
𝑃𝑛 .
Fix 𝑇 ≡ ((𝑇10 , Ψ11 ), (𝑇20 , Ψ12 )), where for each 𝑛, 𝑃𝑛 (𝑇𝑛0 )∩𝑄∗𝑛 nonempty and Ψ1𝑛 is a (possibly
empty) face of Π1𝑛 . Let 𝐴𝑛 (𝑇 ) be the set of points in 𝑃𝑛 (𝑇𝑛0 ) such that the strategies in 𝑇𝑚 are
all best replies. Let 𝑇˜𝑛0 be the unique subset of 𝑇𝑛0 such that the interior of 𝐴𝑛 (𝑇 ) is contained
in the interior of 𝑃𝑛 (𝑇˜𝑛0 ). Let 𝑑˜∗𝑛 (𝑇 ) be the dimension of 𝐴𝑛 (𝑇 ). Since 𝑃𝑚 (𝑇𝑚0 ) contains an
¯∗ .
equilibrium strategy for 𝑚, and likewise for 𝑛, 𝐴𝑛 (𝑇 ) is a subset of 𝐴∗𝑛 (𝑇 ) = 𝑃𝑛 (𝑇˜𝑛0 ) ∩ 𝑄
𝑛
32
SRIHARI GOVINDAN AND ROBERT WILSON
Moreover, since the equilibrium strategy for 𝑚 in 𝑃𝑚 (𝑇𝑚0 ) is undominated, this last set 𝐴∗𝑛 (𝑇 )
is the set 𝑃𝑛 (𝑇˜𝑛0 ) ∩ 𝑄∗𝑛 as well.
Lemma C.6. If 𝑇𝑚1 is nonempty then 𝐴𝑛 (𝑇 ) is a proper face of 𝐴∗𝑛 (𝑇 ).
Proof. This follows from the genericity of payoffs. Fix 𝑡1𝑚 ∈ 𝑇𝑚1 . There exists an information
∗
set ℎ𝑚 ∈ 𝐻𝑚
where it chooses a non-equilibrium action 𝑎. If the path from each (𝑥, 𝑎), for
𝑥 ∈ ℎ𝑚 that is reached under the equilibrium outcome, does not pass through an information
¯ ∗ and 𝐴𝑛 (𝑇 ) would be
set ℎ𝑛 ∈ 𝐻𝑛0 , then 𝑎 would be suboptimal against every equilibrium 𝑄
𝑛
∗
empty. Thus, there exists a first information set ℎ𝑛 ∈ 𝐻𝑛0 and nodes 𝑥 ∈ 𝐻𝑚
and 𝑦 ∈ ℎ𝑛 such
∗
that (𝑥, 𝑎) ≺ 𝑦 and 𝑥 is reached under the equilibrium outcome. Because 𝐴𝑛 (𝑇 ) is nonempty,
there is a strategy 𝑡0𝑛 ∈ 𝑇˜𝑛0 that enables ℎ𝑛 . Clearly, there must be multiple such strategies
that differ in the continuation from ℎ𝑛 , again by genericity. Perturbing the probabilities of
the terminal nodes following 𝑦 does not affect the payoffs to strategies in 𝑇𝑚0 but they affect
the payoff to 𝑡1𝑚 . Hence 𝐴𝑛 (𝑇 ) is a proper face of 𝐴∗𝑛 (𝑇 ).
□
1
Let 𝑇ˆ𝑚1 be the set of strategies 𝑠𝑚 in 𝑆𝑚
(𝑇𝑚0 ) ∖ 𝑇𝑚1 such that: (i) 𝑠𝑚 is an equally good
reply against every point in 𝑃𝑛0 to which the strategies in 𝑇𝑚 are equally good replies. Let
1
𝑇¯𝑚1 be the set of strategies 𝑠𝑚 in 𝑆𝑚
(𝑇𝑚0 ) ∖ (𝑇ˆ𝑚1 ∪ 𝑇𝑚1 ) that are best replies against every point
in 𝐴𝑛 (𝑇 ).
Let 𝐵𝑛∗ (𝑇 ) be the closure of the points in the interior of 𝑃𝑛0 against which the strategies
in 𝑇𝑚 are equally good replies and at least as good replies as strategies in 𝑇¯𝑚1 . Let 𝑑∗𝑛 (𝑇 )
be the dimension of 𝐵𝑛∗ (𝑇 ). If 𝑇𝑚1 is empty, let 𝐵𝑛 (𝑇 ) = 𝑃𝑛0 . Otherwise, let 𝐵𝑛 (𝑇 ) be the
set of points in 𝑃𝑛0 that are of the form 𝜆𝑞𝑛0 + (1 − 𝜆)𝑟𝑛0 such that 𝜆 ⩾ 1, 𝑞𝑛0 ∈ 𝐵𝑛∗ (𝑇 ), and
𝑟𝑛0 ∈ 𝑃𝑛 (𝑇𝑛0 ).
Lemma C.7. Suppose 𝑇𝑚1 is nonempty. 𝐵𝑛 (𝑇 ) is a polyhedron of dimension 𝑑∗𝑛 (𝑇 )+𝑑(𝑇𝑛0 )−
𝑑˜∗𝑛 (𝑇 ). Each maximal face 𝐵𝑛′ (𝑇 ) of 𝐵𝑛 (𝑇 ) satisfies exactly one of the following:
(1) The relative interior of 𝐵𝑛′ (𝑇 ) is contained in the relative interior of a maximal proper
face of 𝑃𝑛0 .
(2) There exists a strategy 𝑟𝑚 ∈ 𝑇¯𝑚1 such that for each 𝑝0𝑛 ∈ 𝐵𝑛′ (𝑇 ), 𝑟𝑚 is an equally
∗
(𝑇 );
good reply against every point of the form 𝜆𝑝0𝑛 + (1 − 𝜆)𝑟𝑛0 , for 0 ⩽ 𝜆 < 1, in 𝐵𝑚
1
1
¯
ˇ
moreover, in this case, letting 𝑅𝑚 be the set of such 𝑟𝑚 , for any 𝑟𝑚 ∈ 𝑅𝑚 , if 𝑟𝑚 is
ˇ 1 is also a best reply
a best reply against a point in 𝐵𝑛∗ (𝑇 ) then every point in 𝑅
𝑚
against this point.
(3) There exists a maximal proper face of 𝑃𝑛 (𝑇𝑛0 ), say 𝑃𝑛 (𝑅𝑛0 ), such that for each 𝑞𝑛 ∈
𝐵𝑛∗ (𝑇 ), 𝑟𝑛 ∈ 𝑃𝑛 (𝑇𝑛0 ) and 𝜆 > 1, if 𝜆𝑞𝑛 + (1 − 𝜆)𝑟𝑛 belongs to 𝐵𝑛′ (𝑇 ), then 𝑟𝑛 belongs
to 𝑃𝑛 (𝑅𝑛0 ); moreover, 𝐴𝑛 (𝑇 ) is contained in 𝑃𝑛 (𝑅𝑛0 ).
AXIOMATIC EQUILIBRIUM SELECTION
33
˜ ∗ (𝑇 ) be the convex cones spanned by 𝑃 0 , 𝑃𝑛 (𝑇 0 ), and 𝐵 ∗ (𝑇 )
Proof. Let 𝑃˜𝑛0 , 𝑃˜𝑛 (𝑇𝑛0 ), and 𝐵
𝑛
𝑛
𝑛
𝑛
0
0
0
0
0
0
0
˜
˜
respectively. Let 𝜉 : 𝑃𝑛 × 𝑃𝑛 (𝑇𝑛 ) → 𝑃𝑛 be the function 𝜉(𝑝𝑛 , 𝑟˜𝑛 ) = 𝑝𝑛 + 𝑟˜𝑛 . Then for
ˆ ) ≡ 𝜉 −1 (𝐵
˜𝑛∗ (𝑇 )) is
each 𝑝˜0𝑛 , 𝜉 −1 (˜
𝑝0𝑛 ) is a set of dimension 𝑑(𝑇𝑛0 ). Hence the dimension of 𝐵(𝑇
ˆ ) is a polyhedron. For each face 𝐵
ˆ ′ (𝑇 ) of 𝐵
ˆ𝑛 (𝑇 ) and for
𝑑(𝑇𝑛0 ) + 𝑑∗𝑛 (𝑇 ) + 1. Obviously 𝐵(𝑇
𝑛
ˆ ′ (𝑇 ) at least one of the following holds: (i) 𝑝0 belongs to the boundary
all points (𝑝0𝑛 , 𝑟˜𝑛0 ) ∈ 𝐵
𝑛
𝑛
of 𝑃𝑛0 ; (ii) there exists a strategy 𝑡𝑚 ∈ 𝑇¯𝑚1 such that 𝜉(𝑝0𝑛 , 𝑟˜𝑛0 ) belongs to the convex cone
spanned by the face of 𝐵𝑛∗ (𝑇 ) where this strategy 𝑡𝑚 is an equally good reply; (iii) 𝑟˜𝑛0 belongs
to the boundary of 𝑃˜𝑛 (𝑇𝑛0 ).
ˆ𝑛 (𝑇 ) onto the first factor. Obviously it is a
Observe now that 𝐵𝑛 (𝑇 ) is the projection of 𝐵
ˆ𝑛 (𝑇 ), (𝑝0 , 𝑟˜0 + 𝜆𝑟0 ) ∈
polyhedron. For each 𝑝0𝑛 ∈ 𝐵𝑛 (𝑇 ) and each 𝑟˜𝑛0 such that (𝑝0𝑛 , 𝑟˜𝑛0 ) ∈ 𝐵
𝑛 𝑛
𝑛
0
∗
0
∗
ˆ
𝐵𝑛 (𝑇 ) for all 𝑟𝑛 ∈ 𝐵𝑛 (𝑇 ) ∩ 𝑃𝑛 (𝑇𝑛 ) and 𝜆 ⩾ 0. If the set 𝐵𝑛 (𝑇 ) is in generic position
(i.e. if the payoffs are in generic position), then for each 𝑝0𝑛 ∈ 𝐵𝑛 (𝑇 ), there exists 𝑟˜𝑛0 such
ˆ𝑛 (𝑇 ) that project to 𝑝0𝑛 can be expressed in the form (𝑝0𝑛 , 𝑟˜𝑛0 + 𝜆𝑟𝑛0 ) for
that all points in 𝐵
some 𝑟𝑛0 ∈ 𝐵𝑛∗ (𝑇 ) ∩ 𝑃𝑛 (𝑇𝑛0 ) and 𝜆 ⩾ 0. Since the set 𝐵𝑛∗ (𝑇 ) ∩ 𝑃𝑛 (𝑇𝑛0 ) is the intersection of
𝑃𝑛 (𝑇𝑛0 ) with the affine space spanned by 𝐴𝑛 (𝑇 ), the dimension of 𝐵𝑛 (𝑇 ) is as asserted. The
enumerated properties of 𝐵𝑛 (𝑇 ) now follow directly from the corresponding points above;
only the last part of property (iii) needs a proof. Suppose 𝑟𝑛0 belongs to a proper face 𝑃𝑛 (𝑅𝑛0 )
ˆ𝑛 (𝑇 ), then so does
of 𝑃𝑛 (𝑇𝑛0 ) and 𝐴𝑛 (𝑇 ) is not contained in 𝑃𝑛 (𝑅𝑛0 ). If (𝑝0𝑛 , 𝜆𝑟𝑛0 ) belongs to 𝐵
(𝑝0𝑛 , 𝜆𝑟𝑛0 + 𝑟¯𝑛0 ) for 𝑟¯𝑛0 ∈ 𝐴𝑛 (𝑇 ) ∖ 𝑃𝑛 (𝑅𝑛0 ) and 𝜆𝑟𝑛0 + 𝑟¯𝑛0 does not belong to the convex cone
generated by 𝑃𝑛 (𝑅𝑛0 ).
□
Let 𝐶𝑛∗ (𝑇 ) be the closure of the set of 𝑞𝑛 in the interior of 𝑃𝑛 such that the strategies in
𝑇𝑚 are all equally good replies and at least as good as strategies in 𝑇ˆ𝑚1 and 𝑆𝑛0 ∖ 𝑇𝑛0 . By
Lemma C.5, the dimension of the face spanned by 𝑇𝑚 is 𝑑(𝑇𝑚0 ) + 𝑑(Ψ1𝑚 ) + 1. By genericity
of payoffs, the dimension of 𝐶𝑛∗ (𝑇 ) is therefore 𝑑(𝑃𝑛 ) − 𝑑(𝑇𝑚0 ) − 𝑑(Ψ1𝑚 ) − 1.
Let 𝐶𝑛 (𝑇 ) be the set of (𝑝1𝑛 , 𝜋𝑛1 ) ∈ 𝑃𝑛1 × Π1𝑛 such that there exist 𝑝0𝑛 ∈ 𝑃𝑛0 , 𝑝2𝑛 ∈ 𝑃𝑛 (𝑇𝑛1 ),
∑
∑
∑
and 𝜇 ∈ ℝ3+ , such that 𝑖 𝜇𝑖 𝑝𝑖𝑛 ∈ 𝐶𝑛∗ (𝑇 ), 𝑖 𝜇𝑖 = 1, 𝜇1 > 0, and 𝜋
¯𝑛1 ( 𝑖 𝜇𝑖 𝑝𝑖𝑛 ) = 𝜋𝑛1 .
Lemma C.8. The set 𝐶𝑛 (𝑇 ) is a polyhedron of dimension 𝑑(𝑃𝑛0 ) + 𝑑(𝑃𝑛1 ) + 𝑑𝑛 (Ψ1𝑛 ) − 𝑑(𝑇𝑚0 ) −
𝑑(Ψ1𝑚 ) − 𝑑∗𝑛 (𝑇 ). On each maximal proper face 𝐶 ′ of 𝐶𝑛 (𝑇 ), exactly one of the following holds
∑
for all (𝑝1𝑛 , 𝜋𝑛1 ) in 𝐶 ′ . If 𝑞𝑛 ∈ 𝐶 ∗ (𝑇 ) is of the form 𝑖 𝜇𝑖 𝑝𝑖𝑛 for some 𝑝0𝑛 ∈ 𝑃𝑛0 and 𝑝2𝑛 ∈ 𝑃𝑛 (𝑇𝑛1 ),
∑
and 𝜋
¯𝑛1 ( 𝑖 𝜇𝑖 𝑝𝑖𝑛 ) = 𝜋𝑛1 , then:
(1) 𝑝1𝑛 belongs to a maximal proper face of 𝑃𝑛1 ;
𝑖
(2) for 𝑖 = 0 or 𝑖 = 1, but not both, there exists 𝑠𝑚 ∈ 𝑆𝑚
∖ 𝑇𝑚𝑖 , which actually belongs to
𝑇ˆ𝑚𝑖 if 𝑖 = 1, that is as good a reply as points in 𝑇𝑚 against 𝑞𝑛 ; moreover, in this case,
if 𝑖 = 0, 𝑇𝑚0 and 𝑠𝑚 span a face of 𝑃𝑚0 of which 𝑃𝑚 (𝑇𝑚0 ) is a maximal proper face; and
¯𝑛1 (𝑠𝑚 ) span a face of Π1𝑚 of which Ψ1𝑚 is a maximal proper face.
if 𝑖 = 1, Ψ1𝑚 and 𝜋
(3) there exists a maximal proper face Ψ′ of Ψ𝑛 such that 𝜋
¯𝑛1 (𝑝2𝑛 ) belongs to Ψ′ .
34
SRIHARI GOVINDAN AND ROBERT WILSON
Proof. We show that 𝐶𝑛 (𝑇 ) is a polyhedron of the stated dimension. Since the construction
is similar to that in the previous lemma, the enumerated properties can be proved just as
before. Let 𝑃𝑛2 be the convex hull of 𝑃𝑛0 and 𝑃𝑛 (𝑇𝑛 ). Using Lemma C.5, the dimension
of 𝑃𝑛2 is 𝑑(𝑃𝑛0 ) + 𝑑(Ψ1𝑛 ) + 1. Let 𝑃˜𝑛2 and 𝑃˜𝑛 be the convex cones spanned by 𝑃𝑛2 and 𝑃𝑛 ,
respectively. Define 𝜉 : 𝑃𝑛1 × 𝑃˜𝑛2 → 𝑃˜𝑛 by 𝜉(𝑝1𝑛 , 𝑝˜2𝑛 ) = 𝑝1𝑛 + 𝑝˜2𝑛 . Then for each 𝑝˜𝑛 in the
interior of 𝑃˜𝑛 , 𝜉 −1 (˜
𝑝𝑛 ) is a set of dimension 𝑑(𝑃𝑛1 ) + 𝑑(𝑃𝑛0 ) + 𝑑(Ψ1𝑛 ) + 1 − 𝑑(𝑃𝑛 ). Letting
𝐶˜𝑛∗ (𝑇 ) be the convex cone spanned by 𝐶𝑛∗ (𝑇 ), the dimension of 𝐶ˆ𝑛 (𝑇 ) ≡ 𝜉 −1 (𝐶˜𝑛∗ (𝑇 )) is
𝑑(𝑃𝑛0 ) + 𝑑(𝑃𝑛1 ) + 𝑑(Ψ1𝑛 ) + 1 − 𝑑(𝑇𝑚0 ) − 𝑑(Ψ1𝑚 ). The function 𝜋
¯𝑛1 extends to 𝑃˜ ∖ { 0 }. 𝐶𝑛 (𝑇 )
is the image of 𝐶ˆ𝑛 (𝑇 ) under the function 𝜒 : 𝑃𝑛1 × 𝑃˜𝑛2 given by 𝜒(𝑝1𝑛 , 𝑝˜2𝑛 ) = (𝑝1𝑛 , 𝜋
¯𝑛1 (𝑝1𝑛 + 𝑝˜2𝑛 ))
and is thus a polyhedron. As will be shown in the course of the proof of the next lemma,
𝐶𝑛∗ (𝑇 ) ∩ 𝑃𝑛2 ⊂ 𝑃𝑛0 . Therefore, 𝐶𝑛∗ (𝑇 ) ∩ 𝑃𝑛2 is the intersection of 𝑃𝑛0 with the affine space
spanned by 𝐵𝑛∗ (𝑇 ). For each (𝑝1𝑛 , 𝑝˜2𝑛 ) ∈ 𝐶ˆ𝑛 (𝑇 ), the point (𝑝1𝑛 , 𝑝˜2𝑛 + 𝜇𝑞𝑛0 ) belongs to 𝐶ˆ𝑛 (𝑇 ) for
all 𝜇 > 0 and 𝑞𝑛0 ∈ 𝐶𝑛∗ (𝑇 ) ∩ 𝑃𝑛2 , and has the same image under 𝜒 as (𝑝1𝑛 , 𝑝˜2𝑛 ). Moreover, if
𝐶𝑛∗ (𝑇 ) is in general position then for each (𝑝1𝑛 , 𝜋𝑛1 ) in 𝐶𝑛 (𝑇 ) every point in its inverse image
under 𝜒 is expressible in this form. Therefore, the dimension of 𝐶𝑛 (𝑇 ) is as given.
□
Let 𝒯 be the collection of 𝑇 ’s such that 𝐴𝑛 (𝑇 ), 𝐵𝑛 (𝑇 ) and 𝐶𝑛 (𝑇 ) are nonempty for each
𝑛. For each 𝑇 ∈ 𝒯 , let 𝒬𝑛 (𝑇 ) = 𝐴𝑛 (𝑇 ) × 𝐵𝑛 (𝑇 ) × 𝐶𝑛 (𝑇 ) for each 𝑛 and let 𝒬(𝑇 ) =
𝒬1 (𝑇 ) × 𝒬2 (𝑇 ).
Lemma C.9. (𝑞 ∗ , 𝑝0 , 𝑝1 , 𝜋 1 ) belongs to 𝒬 iff it belongs to 𝒬(𝑇 ) for some 𝑇 ∈ 𝒯 .
Proof. Suppose for each 𝑛 that 𝑞𝑛∗ ∈ 𝐴𝑛 (𝑇 ), 𝑝0𝑛 ∈ 𝐵𝑛 (𝑇 ), (𝑝1𝑛 , 𝜋𝑛1 ) ∈ 𝐶𝑛 (𝑇 ) for some 𝑇 . Choose
𝑟𝑛0 ∈ 𝑃𝑛 (𝑇𝑛0 ) and 𝜆0𝑛 such that 𝑞𝑛0 ≡ (1 − 𝜆0𝑛 )𝑝0𝑛 + 𝜆0𝑛 𝑟𝑛0 ∈ 𝐵𝑛∗ (𝑇 ). Also, fix 𝑝˜0𝑛 , 𝑟𝑛2 ∈ 𝑃𝑛 (𝑇𝑛1 ),
𝜇0𝑛 , 𝜇1𝑛 , 𝜇2𝑛 such that 𝑞𝑛1 ≡ 𝜇0𝑛 𝑝˜0𝑛 + 𝜇1𝑛 𝑝1𝑛 + 𝜇2𝑛 𝑟𝑛2 belongs to 𝐶𝑛∗ (𝑇 ). Fix points 𝑞¯𝑛0 and 𝑞¯𝑛1
in the interior of 𝐴𝑛 (𝑇 ) and 𝐵𝑛∗ (𝑇 ) for each 𝑛 and consider for each 0 < 𝜀 < 1, the LPS
(𝑞 ∗ , 𝑞˜0 (𝜀), 𝑞˜1 (𝜀)) where for each 𝑛, 𝑞˜𝑛0 (𝜀) = (1−𝜀)¯
𝑞𝑛0 +𝜀𝑞𝑛0 ; and 𝑞˜𝑛1 (𝜀) = (1−𝜀)¯
𝑞𝑛0 +𝜀¯
𝑞𝑛1 (𝜀)+𝜀2 𝑞𝑛1 .
∗
0
1
The strategies in 𝑇𝑛 are equally good replies to 𝑞𝑚
, 𝑞˜𝑚
(𝜀), 𝑞˜𝑚
(𝜀) for all 𝜀. We show that these
∗ 0
1
strategies are lexicographic best replies to (𝑞 , 𝑞˜ (𝜀), 𝑞˜ (𝜀)) for all small 𝜀, which proves that
(𝑞 ∗ , 𝑝0 , 𝑝1 , 𝜋 1 ) belongs to 𝒬.
∗
Observe first that for all 𝜀, a strategy in 𝑆𝑛0 ∖ 𝑇𝑛0 is an equally good reply against 𝑞𝑚
and
1
0
0
𝑞˜𝑚 (𝜀) as strategies in 𝑇𝑛 , by Lemma C.2, and no better a reply against 𝑞˜𝑚 (𝜀) by construction
of 𝐶𝑛∗ (𝑇 ). Now for a strategy 𝑠1𝑛 in 𝑆𝑛1 , consider the strategy 𝑡1𝑛 that agrees with 𝑠1𝑛 at every
information set except that starting at each first information set ℎ𝑛 ∈ 𝐻𝑛0 that 𝑠1𝑛 enables, 𝑡1𝑛
agrees with some 𝑡0𝑛 in 𝑇𝑛0 . Since strategies in 𝑇𝑛0 are at least as good as the other strategies
0
∗
(𝜀), 𝑞˜𝑛1 (𝜀)) as 𝑠1𝑛 . Observe now
, 𝑞˜𝑚
in 𝑆𝑛0 , clearly 𝑡1𝑛 is at least as good a reply against (𝑞𝑚
that 𝑡1𝑛 belongs to 𝑆𝑛1 (𝑇𝑛0 ). If it belongs to 𝑇ˆ𝑛1 , then it is an equally good reply as strategies
1
0
∗
(𝜀) by definition. If it
(𝜀) and no better reply a reply against 𝑞˜𝑚
and 𝑞˜𝑚
in 𝑇𝑛 against 𝑞𝑚
0
∗
1
¯
(𝜀) for all
belongs to 𝑇𝑛 then it is an equally good reply to 𝑞𝑚 , no better reply against 𝑞˜𝑚
AXIOMATIC EQUILIBRIUM SELECTION
35
1
𝜀, and a strictly worse reply against 𝑞˜𝑚
(𝜀) for all small 𝜀, again by construction, since it is
1
an inferior reply against 𝑞¯𝑛 which belongs to the interior of 𝐵𝑛∗ (𝑇 ). Finally, if it belongs to
∗
0
𝑆𝑛1 (𝑇𝑛1 ) ∖ (𝑇ˆ𝑛1 ∪ 𝑇¯𝑛1 ) then it is no better a reply against 𝑞𝑚
and an inferior reply to 𝑞˜𝑚
(𝜀) for
0
all small 𝜀, since it is an inferior reply to 𝑞¯𝑛 by construction of 𝐴𝑛 (𝑇 ). Thus the strategies
in 𝑇 are lexicographic best replies to (𝑞 ∗ , 𝑞˜0 (𝜀), 𝑞˜1 (𝜀)) for all small 𝜀.
Before proceeding to prove the converse, we use the above argument to show that the
intersection of 𝐶𝑛∗ (𝑇 ) with the convex hull 𝑃𝑛2 of 𝑃𝑛0 with 𝑃𝑛 (𝑇𝑛 ) is in fact the intersection
𝐹 of 𝑃𝑛0 with the affine space spanned by 𝐵𝑛∗ (𝑇 )—a fact that was asserted, but not proved,
in the course of the proof of the previous lemma. Take a point 𝑞𝑛1 in 𝐶𝑛∗ (𝑇 ) ∩ 𝑃 2 . If it
belongs to 𝑃𝑛0 , then in fact it belongs to 𝐹 by the definitions of 𝐵𝑛∗ (𝑇 ) and 𝑇ˆ𝑚1 . If it does
not belong to 𝑃𝑛0 , then it assigns a positive weight to some strategy 𝑠1𝑛 ∈ 𝑇𝑛1 . The above
∗
argument applied when using this 𝑞𝑛1 shows that 𝑞𝑚
is a best reply to (𝑞𝑛∗ , 𝑞˜𝑛0 (𝜀), 𝑞˜𝑛1 (𝜀)) and the
∗
strategies in 𝑇𝑛 and 𝑆𝑛0 are best replies to 𝑞𝑚
. Observe now that 𝑞˜𝑛1 (𝜀) is a convex combination
∗
of strategies in 𝑆𝑛0 and 𝑇𝑛1 . Therefore, for all small 𝛿, ((1 − 𝛿 − 𝛿 2 )𝑞𝑛∗ + 𝛿 𝑞˜𝑛0 (𝜀) + 𝛿 2 𝑞˜𝑛1 (𝜀), 𝑞𝑚
)
is an equilibrium if 𝜀 is small as well. But these points induce different outcomes because
𝑞˜𝑛1 (𝜀) has a non-equilibrium strategy, namely one in 𝑇𝑛1 , in its support, which is impossible.
Thus, 𝐶𝑛∗ (𝑇 ) ∩ 𝑃𝑛2 = 𝐹 as claimed.
Returning to the proof of this lemma, suppose (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) belongs to 𝒬. Let 𝑞𝑛0 =
(1 − 𝜆0𝑛 )𝑝0𝑛 + 𝜆0𝑛 𝑟𝑛0 and let 𝑞𝑛1 = 𝜇0𝑛 𝑝˜0𝑛 + 𝜇1𝑛 𝑝1𝑛 + 𝜇2𝑛 𝑟𝑛2 where (1 − 𝜆0𝑛 )𝑞𝑛∗ + 𝜆0𝑛 𝑟𝑛0 is a best reply
against (𝑞 ∗ , 𝑞 0 , 𝑞 1 ); and 𝑟𝑛2 , if 𝜇2𝑛 > 0, is a best reply against 𝑞𝑛∗ and a weakly better reply
against (𝑞 ∗ , 𝑞 0 , 𝑞 1 ) than all the strategies in 𝑃𝑛1 . Let 𝑄0𝑛 be the face of 𝑃𝑛0 that contains
(1 − 𝜆0𝑛 )𝑞𝑛∗ + 𝜆0𝑛 𝑟𝑛0 in its interior. Let 𝑄1𝑛 be the face of 𝑃𝑛1 that contains 𝑟𝑛2 in its interior if
𝜇2𝑛 > 0. Let 𝑇𝑛0 be the set of strategies 𝑡𝑛 in 𝑆𝑛0 such that if 𝑡𝑛 enables a first information set
ℎ𝑛 ∈ 𝐻𝑛0 then the choices from there on prescribed by 𝑡0𝑛 coincide with the choices dictated
by some vertex of 𝑄0𝑛 or 𝑄1𝑛 that enables ℎ𝑛 . Observe that each 𝑡0𝑛 ∈ 𝑇𝑛0 is optimal against
(𝑞 ∗ , 𝑞 0 , 𝑞 1 ). If 𝜇2𝑛 > 0, let Ψ1𝑛 be the face of Π1𝑛 that contains 𝜋
¯𝑛1 (𝑟𝑛2 ) in its interior; otherwise
let Ψ1𝑛 be the empty set.
We can now assume without loss of generality that the strategies in 𝑄0𝑛 and 𝑄1𝑛 are equally
∗
0
1
good replies against (𝑞𝑚
, 𝑞𝑚
, 𝑞𝑚
) and hence best replies. Indeed, if for 𝑖 either 0 or 1, if the
1
𝑖
𝑖
strategies in 𝑄𝑛 do not yield the same payoff against 𝑞𝑚
as those in 𝑄0𝑛 , modify 𝑞𝑚
as follows:
∗
0
˜ 0𝑚 of 𝑄0𝑚 containing 𝑞𝑚
in its interior such that the strategies
in the face 𝑄
pick a point 𝑟ˇ𝑚
1
0
in 𝑄𝑛 are equally good replies, the strategies in 𝑄𝑛 do strictly better than the strategies in
𝑖
∈ [0, 1] such
𝑄0𝑛 and at least as well as the other strategies in 𝑆𝑛1 . There exists a unique 𝜈𝑚
𝑖 𝑖
𝑖
𝑖
1
0
𝑟ˇ𝑚 .
+ 𝜈𝑚
that the strategies in 𝑄𝑛 and 𝑄𝑛 are now equally good replies against (1 − 𝜈𝑚 )𝑞𝑚
Thus, our assumption is without loss of generality.
Since the strategies in 𝑄0𝑛 and 𝑄1𝑛 are best replies. There remains to show that every
1
0
∗
). Fix 𝑠𝑛 ∈ 𝑆𝑛 (𝑇𝑛0 ; Ψ1𝑛 ). To show
, 𝑞𝑚
, 𝑞𝑚
strategy in 𝑆𝑛 (𝑇𝑛0 ; Ψ1𝑛 ) is a best reply against (𝑞𝑚
36
SRIHARI GOVINDAN AND ROBERT WILSON
that it is a best reply it is sufficient to show that an information set ℎ𝑛 that is enabled by 𝑠𝑛
is enabled by some vertex of either 𝑄0𝑛 or 𝑄1𝑛 and that this vertex agrees with 𝑠𝑛 ’s choice 𝑎𝑛
there. Suppose that this ℎ𝑛 is in 𝐻𝑛∗ and 𝑎𝑛 ∈ 𝐴∗𝑛 , or ℎ𝑛 belongs to 𝐻𝑛0 ; then obviously some
strategy in 𝑄0𝑛 or 𝑄1𝑛 enables ℎ𝑛 and chooses 𝑎𝑛 , by the definition of 𝑇𝑛0 . If ℎ𝑛 ∈ 𝐻𝑛∗ and
𝑎𝑛 ∈
/ 𝐴∗𝑛 or ℎ𝑛 follows some information set in 𝐻𝑛∗ by the choice of a non-equilibrium action,
then some strategy in 𝑄1𝑛 enables it and chooses this action, since otherwise 𝑠𝑛 enables a
terminal node that is excluded by all strategies in 𝑄1𝑛 , contradicting the assumption that
𝜋
¯𝑛1 (𝑠𝑛 ) ∈ Ψ1𝑛 . Thus 𝑠𝑛 is a best reply and (𝑞 ∗ , 𝑞 0 , 𝑞 1 ) belongs to 𝒬(𝑇 ).
□
Lemma C.10. 𝒬 is a pseudomanifold of dimension 𝑑ˆ ≡ 𝑑(ℙ).
ˆ By the previous lemma 𝒬 = ∪𝑇 ∈𝒯 𝒬(𝑇 ).
Proof. Each 𝒬(𝑇 ) is a polyhedron of dimension 𝑑.
ˆ To show that 𝒬 is a pseudomanifold, we establish three
Therefore, 𝒬 has dimension 𝑑.
facts for each (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) that belongs to some 𝒬(𝑇 ): (1) if (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) belongs to
the interior of 𝒬(𝑇 ), then it does not belong to the interior of 𝒬(𝑅) for 𝑅 ∕= 𝑇 ; (2) if
(𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) is a generic point in a maximal proper face 𝒬′ of 𝒬(𝑇 ), then it does not
belong to 𝒬(𝑅) for any 𝑅 ∕= 𝑇 if (𝑝0 , 𝑝1 ) ∈ ∂ℙ, and it belongs to the boundary of 𝒬(𝑅) for
exactly one other 𝑅 ∕= 𝑇 if (𝑝0 , 𝑝1 ) ∈
/ ∂ℙ; moreover in the latter case it belongs to the interior
of a maximal proper face of this 𝒬(𝑅) as well; (3) given 𝑇, 𝑅 ∈ 𝒯 , there exists a finite chain
𝑇 = 𝑇 (0), . . . , 𝑇 (𝑘) = 𝑅 such that for each 0 ≤ 𝑗 ≤ 𝑘 − 1, 𝒬(𝑇 (𝑗)) ∩ 𝒬(𝑇 (𝑗 + 1)) is a subset
of a maximal proper face of each and has a nonempty interior in this face.
Fix 𝑇 = (𝑇 0 , Ψ1 ) and 𝑥 = (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) ∈ 𝒬(𝑇 ). For each 𝑛, choose 𝑞𝑛0 ≡ (1 − 𝜆0𝑛 )𝑝0𝑛 +
𝜆0𝑛 𝑟𝑛0 in 𝐵𝑛∗ (𝑇 ) and 𝑞𝑛1 ≡ 𝜇0𝑛 𝑝˜0𝑛 + 𝜇1𝑛 𝑝1𝑛 + 𝜇2𝑛 𝑟𝑛1 ∈ 𝐶𝑛∗ (𝑇 ).
We start with (1). Suppose now that 𝑥 belongs to the interior of 𝒬(𝑅) for some 𝑅. We
show that 𝑅 = 𝑇 . Since 𝑥 belongs to the interior of 𝒬(𝑇 ), we can assume that every strategy
0
0
in 𝑆𝑚
∖ 𝑇𝑚0 is inferior to 𝑞𝑛1 . Let 𝑠𝑚 be a strategy in 𝑆𝑚
∖ 𝑇𝑚0 . Since 𝑠𝑚 is an inferior reply
1
against 𝑞𝑚
compared to the strategies in 𝑇𝑛0 , by Lemma C.3 there exists an information set
ℎ𝑛 ∈ 𝐻𝑛0 that is enabled by 𝑠𝑛 where the action chosen by 𝑠𝑛 is suboptimal and different
from the action chosen by every 𝑡𝑛 ∈ 𝑇𝑛0 that enables ℎ𝑛 . But the posterior belief over the
1
1
terminal nodes following ℎ𝑛 computed from 𝑞𝑚
can be computed from 𝜋𝑚
. This implies that
′
1
′
1
1
for any 𝑞𝑚 such that 𝜋
¯𝑚 (𝑞𝑚 ) = 𝜋𝑚 , 𝑠𝑛 is an inferior strategy. Therefore, (𝑝1𝑚 , 𝜋𝑚
) cannot
0
0
0
0
belong to 𝐶𝑚 (𝑅) unless 𝑅𝑚 ⊆ 𝑇𝑚 . Moreover, if 𝑅𝑚 ⊊ 𝑇𝑚 , then it cannot belong to the
0
are also optimal. Thus, if 𝑥 belongs to the
interior of 𝐶𝑚 (𝑅), since strategies in 𝑇𝑚0 ∖ 𝑅𝑚
1
0
0
interior of 𝒬(𝑅), 𝑅𝑚 = 𝑇𝑚 for each 𝑚. If Ψ𝑚 is empty, this implies that 𝑅𝑚 = 𝑇𝑚 . Suppose
1
1
and
(𝑝1𝑚 ) ∕= 𝜋𝑚
¯𝑚
now that Ψ1𝑚 is nonempty. Since 𝑥 is in the interior we can assume that 𝜋
1 1
1
1
¯𝑛 (𝑟𝑚 ) can be computed uniquely from 𝑝1𝑚
that 𝜋
¯𝑛 (𝑟𝑚 ) is in the interior of Ψ𝑚 . Observe that 𝜋
1
1
1
and computing the boundary
(𝑝1𝑚 ) through 𝜋𝑚
by taking the line segment from 𝜋
¯𝑚
and 𝜋𝑚
point of this line. This implies that if 𝑥 belongs to the interior of 𝒬(𝑅), then 𝑅𝑚 = (𝑇𝑚0 , Ψ1𝑚 ).
Thus 𝑅 = 𝑇 .
AXIOMATIC EQUILIBRIUM SELECTION
37
We turn to point (2). Suppose now 𝑥 belongs to the relative interior of a maximal proper
face of 𝒬(𝑇 ) and that (𝑝0 , 𝑝1 ) belongs to ∂ℙ. Then if it belongs to another 𝒬(𝑅) it cannot be
in the interior and must belong to the boundary. The arguments of the previous paragraph
apply to show that 𝑥 does not belong to the interior of a maximal face of 𝒬(𝑅): indeed,
0
it relied on strategies in 𝑆𝑚
∖ 𝑇𝑚0 being inferior to 𝑞𝑛1 , and 𝑞𝑛0 (resp. 𝑞𝑛1 ) not belonging to
the boundary of 𝐵𝑛 (𝑇 ) (resp. 𝐶𝑛 (𝑇 )). Thus 𝑥 must belong to a face of dimension at most
𝑑ˆ − 2. The set of such points in this maximal proper face of 𝒬(𝑇 ) then has dimension at
most 𝑑ˆ − 2, i.e. it is nongeneric.
Suppose that 𝑥 belongs to the relative interior of a maximal proper face of 𝒬(𝑇 ) but
that (𝑝0 , 𝑝1 ) is in the interior in ℙ. Then for exactly one 𝑛, just one of the following hold:
(2a) 𝑞𝑛∗ belongs to the boundary of 𝐴𝑛 (𝑇 ); (2b) 𝑝0𝑛 belongs to the boundary of 𝐵𝑛 (𝑇 );
(2c) (𝑝1𝑛 , 𝜋𝑛1 ) belongs to the boundary of 𝐶𝑛 (𝑇 ). We start with (2c). By the properties we
proved for 𝐶𝑛 (𝑇 ) in Lemma C.8, and since (𝑝0 , 𝑝1 ) ∈
/ ∂ℙ, either property (ii) or property
(iii) of that lemma holds. Under property (ii) 𝑥 belongs to the boundary of 𝒬(𝑅) where
0
𝑖
𝑅 = ((𝑅𝑚
, Φ1𝑚 ), (𝑇𝑛 , Ψ1𝑛 )) is defined as follows. If the strategy 𝑟𝑚
identified there belongs
0
0
𝑖
0
to 𝑆𝑚 , then 𝑅𝑚 is the vertex set of the face spanned by 𝑟𝑚 and 𝑇𝑚 , while Φ1𝑚 = Ψ1𝑛 ; if the
𝑖
1
0
strategy 𝑟𝑚
belongs to 𝑆𝑚
(𝑇𝑚0 ; Ψ1𝑚 ), then 𝑅𝑚
= 𝑇𝑚0 and Φ1𝑚 is a face of Π1𝑚 that has Ψ1𝑚 as
1
1
a maximal face with 𝜋𝑚
(𝑟𝑚
) ∈ Φ1𝑚 ∖ Ψ1𝑚 . Under property (iii) 𝑥 belongs to the boundary of
𝒬(𝑅) where 𝑅 = ((𝑇𝑛0 , Φ1𝑛 ), (𝑇𝑚0 , Ψ1𝑚 )) where Φ1𝑛 is the maximal proper face of Ψ1𝑛 identified
there.
Suppose 𝑥 satisfies (2b). Then by the properties we proved for 𝐵𝑛 (𝑇 ) in Lemma C.7,
˜ 1 be the
either property (ii) or property (iii) of that lemma holds. Under property (ii) let 𝑅
𝑚
1
0
1
˜
set of strategies in 𝑇𝑚 that are now best replies against 𝑞𝑛 . Let Φ̃𝑚 be the smallest face of
1
1
1
1
˜𝑚
Π1𝑚 that contains Ψ1𝑚 and the vectors 𝜋𝑚
(˜
𝑟𝑚
) for 𝑟˜𝑚
∈𝑅
. Then the strategies in 𝑇𝑚0 and
1
𝑆𝑚
(Φ1𝑚 ) are equally good replies against 𝑞𝑛0 . Moreover by the genericity of 𝑥, if one of these
strategies is a best reply against a point in 𝐵𝑛∗ (𝑇 ) then all these points are best replies as
well. For each face Φ1𝑚 of Φ̃1𝑚 that has Ψ1𝑚 a maximal proper face, choose a strategy 𝑟𝑚 (Φ1𝑚 )
that maps to a vertex of Φ1𝑚 that is not contained in Ψ1𝑚 . The set of points in 𝐵𝑛∗ (𝑇 ) against
˜ 𝑚 are as good replies as 𝑇𝑚 has dimension 𝑑∗𝑛 (𝑇 ) − 1. However, the
which the strategies in 𝑅
set of points in 𝐶𝑛∗ (𝑇 ) where two or more of these strategies 𝑟𝑚 (Φ1𝑚 ) are also best replies
has dimension 𝑑(𝑃𝑛 ) − 𝑑(𝑇𝑚0 ) − 𝑑(Ψ1𝑚 ) − 3 or less. Therefore, for 𝑅 and 𝑅′ of the form
((𝑇𝑛0 , Ψ1𝑛 ), (𝑇𝑚0 , Φ1𝑚 )) the set of (𝑝1𝑛 , 𝜋𝑛1 ) that lies in the intersection 𝐶𝑛 (𝑇 ) ∩ 𝐶𝑛 (𝑅) ∩ 𝐶𝑛 (𝑅′ )
is at most 𝑑0𝑛 + 𝑑1𝑛 + 𝑑𝑛 (Ψ1𝑛 ) − 𝑑∗𝑛 (𝑇 ) − 1 or less. This implies that generic (𝑝1𝑛 , 𝜋𝑛1 ) in 𝐶𝑛 (𝑇 )
belongs to at most one of these sets. Moreover, if 𝑥 belongs to 𝒬(𝑅), then it belongs to
1
is a convex combination of
the boundary of 𝒬(𝑅): indeed the point 𝜙1𝑚 in Φ1𝑚 such that 𝜋𝑚
1
1
1
𝜋
¯𝑚 (𝑝𝑚 ) and 𝜙𝑚 is uniquely determined, as we argued above; since this point belongs to Ψ1𝑚 ,
which is a face of Φ1𝑚 , 𝑥 indeed belongs to the boundary of 𝒬(𝑅) if it belongs to 𝒬(𝑅). To
38
SRIHARI GOVINDAN AND ROBERT WILSON
1
finish the proof of this case, we now show that 𝑥 belongs to at least one 𝒬(𝑅). Take an 𝑟˜𝑚
1
˜𝑚
that yields the highest payoff against 𝑞𝑛1 among the strategies in 𝑅
. If this payoff is higher
0
0
than the payoff to the strategies in 𝑇𝑚 , pick a point 𝑞¯𝑛 in the interior of 𝐵𝑛∗ (𝑇 ) and replace
1
𝑞𝑛1 with 𝑞˜𝑛1 (𝜀) ≡ (1 − 𝜀)𝑞𝑛1 + 𝜀¯
𝑞𝑛0 where 𝜀 is the unique number where the strategies 𝑇𝑚0 and 𝑟˜𝑚
1
1
are equally good replies; then 𝑥 belongs to some 𝒬(𝑅) that has 𝜋𝑚
(˜
𝑟𝑚
) as an extra vertex.
1
0
0
If the payoff to 𝑟˜𝑚 is lower, take a point 𝑞¯𝑛 in 𝑃𝑛 against which the strategies in 𝑇𝑚0 are
˜ 1 and repeat the argument to show that 𝑥
equally good and worse than the strategies in 𝑅
𝑚
belongs to 𝒬(𝑅).
Finally, suppose that 𝑥 satisfies (2a). Let 𝐴′𝑛 be a maximal proper face of 𝐴𝑛 (𝑇 ) that
1
1
ˇ𝑚
contains 𝑞𝑛∗ in its interior. Let 𝑅
be the set of strategies 𝑟ˇ𝑚 in 𝑆𝑚
(𝑇𝑚0 ) ∖ (𝑇𝑚1 ∪ 𝑇ˆ𝑚1 ∪ 𝑇¯𝑚1 )
1
ˇ𝑚
that are best replies against all the points in 𝐴′𝑛 . Observe that 𝑅
is nonempty if 𝑞𝑛∗ belongs
to the interior of 𝑃𝑛 (𝑇˜𝑛0 ). Indeed, in this case, the interior of 𝐴′𝑛 which is a face of 𝐴𝑛 (𝑇 ) is
1
contained in the interior of 𝑃𝑛 (𝑇˜𝑛0 ), which implies that some strategy in 𝑆𝑚
is now optimal
0
0
0
against every point in this face. Let ℛ𝑛 be the set of subsets 𝑅𝑛 of 𝑇𝑛 such 𝑃𝑛 (𝑅𝑛0 ) is a
maximal proper face of 𝑃𝑛 (𝑇𝑛0 ) and 𝑃𝑛 (𝑅𝑛0 ) ∩ 𝑄∗𝑛 = 𝐴′𝑛 . Observe that ℛ0𝑛 is nonempty if 𝑞𝑛∗
belongs to the boundary of 𝑃𝑛 (𝑇˜𝑛0 ).
1
1
1
1
ˇ𝑚
Let Φ̌1𝑚 be the smallest face of Π1𝑚 that contains Ψ1𝑚 and the vectors 𝜋𝑚
(ˇ
𝑟𝑚
) for 𝑟ˇ𝑚
∈𝑅
.
1
1
1
1
1
For each face Φ𝑚 of Φ̌𝑚 that has Ψ𝑚 as a maximal proper face, choose a strategy 𝑟ˇ𝑚 (Φ𝑚 )
that maps to a vertex of Φ1𝑚 that is not contained in Ψ1𝑚 . Take 𝑅 and 𝑅′ of the form
((𝑇𝑛0 , Ψ1𝑛 ), (𝑇𝑚0 , Φ1𝑚 )) where Φ1𝑚 has Ψ1𝑚 as a maximal face. Since 𝐴′𝑛 is a face of 𝐴𝑛 (𝑇 ),
ˇ 1 are inferior replies to points in the
𝑑˜∗𝑛 (𝑅) = 𝑑˜∗𝑛 (𝑅′ ) = 𝑑˜∗𝑛 (𝑇 ) − 1. Since the strategies in 𝑅
𝑚
∗
∗
interior of 𝐴𝑛 (𝑇 ), 𝐵𝑛 (𝑅) if nonempty has dimension 𝑑𝑛 (𝑇 )−1. Therefore, if 𝐵𝑛 (𝑅) ∕= 𝐵𝑛 (𝑅′ ),
their intersection with 𝐵𝑛 (𝑇 ) has codimension 1 in 𝐵𝑛 (𝑇 ) and a generic 𝑥 cannot belong to
two of these sets at once. On the other hand, if 𝐵𝑛 (𝑅) = 𝐵𝑛 (𝑅′ ) then an argument similar to
that under case (2b) shows that generic (𝑝1𝑛 , 𝜋𝑛1 ) ∈ 𝐶𝑛 (𝑇 ) can belong to at most one of these
sets, 𝐶𝑛 (𝑅) and 𝐶𝑛 (𝑅′ ). Hence a generic 𝑥 belongs to at most one of these sets. Likewise, for
𝑅 of the form ((𝑅𝑛0 , Ψ1𝑛 ), (𝑇𝑚0 , Ψ1𝑚 )) with 𝑅𝑛0 ∈ ℛ0𝑛 and 𝑅′ of the form ((𝑅𝑛′ 0 , Ψ1𝑛 ), (𝑇𝑚0 , Ψ1𝑚 ))
or ((𝑇𝑛0 , Ψ1𝑛 ), (𝑇𝑚0 , Φ1𝑚 )), the intersection of 𝐵𝑛 (𝑇 ) with 𝐵𝑛 (𝑅) and 𝐵𝑛 (𝑅′ ) has codimension
at least one. Thus 𝑥 belongs to at most one set 𝒬(𝑅). To finish the proof of this part, we
show that it belongs to at least one such set.
1
(Φ1𝑚 ) be
Suppose that the interior of 𝐴′𝑛 is contained in the interior of 𝑃𝑛 (𝑇˜𝑛0 ). Let 𝑟ˇ𝑚
a strategy that is a lexicographic best reply to (𝑞𝑛0 , 𝑞𝑛1 ) among the strategies in this class.
1
(Φ1𝑚 ) is a lexicographic weakly better (resp. strictly inferior) reply against (𝑞𝑛0 , 𝑞𝑛1 ) we
If 𝑟ˇ𝑚
1
(Φ1𝑚 ) is at least as
choose a point 𝑞¯𝑛0 in the interior of 𝑃𝑛 (𝑇˜𝑛0 ) against which the strategy 𝑟ˇ𝑚
good as the other points in this class and inferior (resp. superior) to strategies in 𝑇𝑚0 . The
strategies in 𝑇𝑚0 and 𝑆𝑚 (𝑇𝑚0 ; Φ1𝑚 ) are now equally good replies against some average of 𝑞𝑛0 and
𝑞¯𝑛0 , as well as some average of 𝑞𝑛1 and 𝑞¯𝑛0 . The point 𝑥 then belongs to ((𝑇𝑚0 , Φ1𝑚 ), (𝑇𝑛0 , Ψ1𝑛 )).
AXIOMATIC EQUILIBRIUM SELECTION
39
Suppose now that 𝑞𝑛∗ belongs to the boundary of 𝑃𝑛 (𝑇˜𝑛0 ), then ℛ0𝑛 is nonempty. There
ˇ 0 in ℛ0 and a point 𝑞˜0 ∈ 𝐵𝑛 (𝑇 ) that is a convex combination of 𝑝0 and some point
exists 𝑅
𝑛
𝑛
𝑛
𝑛
1
ˇ 𝑛0 ). If the strategies in 𝑅
ˇ𝑚
in 𝑃𝑛 (𝑅
are weakly inferior against 𝑞˜𝑛0 to the strategies in 𝑇𝑚 ,
ˇ 𝑛0 , Ψ1𝑛 ), (𝑇𝑚0 , Ψ1𝑚 )). Otherwise, as above, we can replace 𝑞˜𝑛0 by a convex
then 𝑥 belongs 𝑄((𝑅
combination 𝑞¯𝑛0 of 𝑞˜𝑛0 with a point in the interior of 𝐴𝑛 (𝑇 ) and replace 𝑞𝑛1 with an a point
𝑞¯𝑛1 that is a convex combination of 𝑞𝑛1 with either 𝑞˜𝑛0 or a point in the interior of 𝐴𝑛 (𝑇 )
1
depending on whether the strategy 𝑟ˇ𝑚
(Φ1𝑚 ) that is superior to the strategies in 𝑇𝑚 against
𝑞˜𝑛0 is inferior or weakly superior in comparison against 𝑞𝑛1 . The points 𝑞¯𝑛0 and 𝑞¯𝑛1 belong to
𝐵𝑛 (𝑇 ′ ) and 𝐶𝑛 (𝑇 ′ ) respectively, where 𝑇 ′ = ((𝑇𝑛 , Ψ1𝑛 ), (𝑇𝑚 , Φ1𝑚 ) and thus 𝑥 belongs to 𝑄(𝑇 ′ ).
We turn now to (3). Given 𝒬(𝑇 ) and 𝒬(𝑅) for 𝑇 = (𝑇 0 , Ψ0 ) ∕= 𝑅 = (𝑅0 , Φ1 ), we will first
construct a sequence 𝑇 = 𝑇 (1), . . . , 𝑇 (𝑘) = 𝑇˜ where 𝑇˜ = ((𝑇˜𝑛0 , ∅), (𝑇˜𝑚0 , ∅)). And, likewise
˜ Then we will show how to construct a sequence from 𝑇˜ to 𝑅.
˜
one from 𝑅 to 𝑅.
In case 𝑇˜𝑛0 ∕= 𝑇𝑛0 for some 𝑛, let 𝑇 = 𝑇 (0), . . . , 𝑇 (𝑘) be a sequence where for each 𝑗 > 0,
𝑇 (𝑗) = ((𝑇𝑛0 (𝑗), Ψ1𝑛 ), (𝑇𝑚0 (𝑗), Ψ1𝑛 )) with 𝑃𝑛 (𝑇𝑛0 (𝑗))×𝑃𝑚 (𝑇𝑚0 (𝑗)) being a maximal proper face of
𝑃𝑛 (𝑇𝑛0 (𝑗−1))×𝑃𝑚 (𝑇𝑚0 (𝑗−1)), and 𝑇𝑛 (𝑘) = 𝑇˜𝑛0 for each 𝑛. This sequence generates a sequence
of polyhedra 𝒬(𝑇 ) = 𝒬(𝑇 (0)), . . . , 𝒬(𝑇 (𝑘)) where for each 𝑗 > 0, the intersection of 𝒬(𝑇 (𝑗))
with 𝒬(𝑇 (𝑗 − 1)) is contained in a maximal proper face of each and has a nonempty interior.
After this operation we have 𝒬(𝑇˜(𝑘)), where 𝑇˜(𝑘) = ((𝑇˜𝑛0 , Ψ1𝑛 ), (𝑇˜𝑚0 , Ψ1𝑚 )). In case Ψ1𝑛
is nonempty for some 𝑛, let Ψ1 = (Ψ1𝑛 (0), Ψ1𝑚 (0)), . . . , (Ψ1𝑛 (𝑙), Ψ1𝑚 (𝑙)) = (∅, ∅) be a sequence
such that for each 1 ⩽ 𝑗 ≤ 𝑙, Ψ1𝑚 (𝑗)×Ψ1𝑚 (𝑗) is a maximal proper face of Ψ1𝑛 (𝑗 −1)×Ψ1𝑚 (𝑗 −1),
and Ψ1 (𝑙) = ∅. This way we can connect 𝒬(𝑇˜(𝑘)) with 𝒬(𝑇˜) where 𝑇˜ = ((𝑇˜𝑛0 , ∅), (𝑇˜𝑚0 , ∅)).
˜ for two sets 𝑇, 𝑅 in 𝒯 . Because 𝑄∗𝑛 is connected for
Now we show how to connect 𝑇˜ with 𝑅
0
0
˜ 0 where for each
each 𝑛, there exists a sequence 𝑇˜0 = (𝑆𝑛0 (0), 𝑆𝑚
(0)), . . . , (𝑆𝑛0 (𝑙), 𝑆𝑚
(𝑙)) = 𝑅
𝑛
0
1 ⩽ 𝑗 ⩽ 𝑙, either 𝑃𝑛 (𝑆𝑛0 (𝑗)) × 𝑃𝑛 (𝑆𝑚
(𝑗)) is either a maximal proper face of 𝑃𝑛 (𝑆𝑛0 (𝑗 + 1))
or vice versa and for each 𝑛, 𝑄∗𝑛 intersects the interior of the set 𝑃𝑛 (𝑆𝑛0 (𝑗)). This generates
0
a sequence 𝒬0 , . . . 𝒬𝑗 , where 𝒬𝑗 = ((𝑆𝑛0 (𝑗), ∅), (𝑆𝑚
(𝑗), ∅)). Thus we have constructed a
sequence of sets in 𝒯 that connect 𝑇 and 𝑅.
□
This concludes the proof of the first statement in the theorem. Next we prove the second
statement, invoking now the original definition of a stable set in Mertens [22].
Lemma C.11. 𝑄∗ is a stable set if and only if the projection map Ψ : (𝒬, ∂𝒬) → (ℙ, ∂ℙ) is
essential.
Proof. Let 𝑌 = [0, 1] × 𝑃 . For each 0 < 𝜀 ⩽ 1, let 𝑌𝜀 = [0, 𝜀] × 𝑃 and let ∂𝑌𝜀 be the
boundary of 𝑌𝜀 . Each (𝜀, 𝑝) ∈ 𝑌 defines a strategic game 𝐺(𝜀, 𝑝) where the strategy set is 𝑃
but where the payoff from an enabling strategy profile 𝑞 is the payoff in 𝐺 from the profile
(1 − 𝜀)𝑞 + 𝜀𝑝. If 𝑞 is an equilibrium of 𝐺(𝜀, 𝑝), we say that (1 − 𝜀)𝑞 + 𝜀𝑝 is a perturbed
equilibrium of 𝐺(𝜀, 𝑝). Let ℰ be the closure of the set of (𝜀, 𝑝, 𝑞) such that (𝜀, 𝑝) ∈ 𝑌1 ∖ ∂𝑌1
40
SRIHARI GOVINDAN AND ROBERT WILSON
and 𝑞 is a perturbed equilibrium of 𝐺(𝜀, 𝑝). Let 𝜃 be the projection map from ℰ. For each
subset 𝐸 of ℰ and each 0 < 𝜀, let (𝐸𝜀 , ∂𝐸𝜀 ) be 𝐸 ∩ 𝜃−1 (𝑃𝜀 , ∂𝑃𝜀 ).
In [11] we show that there exists 0 < 𝜀¯ ⩽ 1 and a finite number of subsets 𝐸 1 , . . . , 𝐸 𝐾
of ℰ such that for each 0 < 𝜀 ⩽ 𝜀¯: (i) (𝐸𝜀𝑘 , ∂𝐸𝜀𝑘 ) is a pseudomanifold (in fact an orientable
semi-algebraic homology manifold) of dimension 𝑑(𝑃 ) + 1 for each 𝑘; (ii) 𝐸𝜀𝑘 ∩ 𝐸𝜀𝑗 ⊂ 𝜃−1 (∂𝑃1 )
for 𝑘 ∕= 𝑗; (iii) ∪𝑘 𝐸𝜀𝑘 = ℰ𝜀 . We will assume that 𝜀¯ is small enough such that for each player
𝑛, and each (𝜀, 𝑝, 𝑞) ∈ 𝐸𝜀¯, if a strategy 𝑠𝑛 is optimal against a strategy 𝑞 in Γ, then it is
¯ ∗ , the component of equilibria containing 𝑄∗ .
optimal against some point in 𝑄
𝑛
𝑛
One could define the set of perturbations for the normal form of the game and consider the
graph of the equilibria over this space. In [7] we show that there exists a neighborhood of Σ∗
that is disjoint from the other components of Γ and an 𝜀 > 0 such that the set of 𝜀-perfect
equilibria in this neighborhood (viewed as points in the graph of equilibria) is connected. The
corresponding set of 𝜀-perfect equilibria in enabling strategies is therefore connected. Thus
there exists some 𝑘 such that 𝐸0𝑘 = { 0 }×𝑄∗ and for each 𝑗 ∕= 𝑘, 𝐸0𝑗 ∩({ 0 }×𝑄∗ ) is empty. For
simplicity in notation we refer to this 𝐸 𝑘 as simply 𝐸. According to Mertens’ [22] definition,
𝑄∗ is stable iff the projection 𝜃 from 𝐸𝜀 to 𝑌𝜀 is cohomologically essential for some (and
then all smaller) 0 < 𝜀 ⩽ 𝜀¯. Moreover, since 𝐸𝜀 is a pseudomanifold, 𝜃 is cohomologically
essential iff it is essential in homotopy [23, Theorem, Section 4E]. By [11, Lemmas A.3, A.4],
this is equivalent to saying that Ψ is essential in the sense we have used it in Section 5.
It is now sufficient to prove that Ψ is essential iff the projection 𝜃 from 𝐸𝜀 to 𝑌𝜀 is essential
for all small 𝜀. For each 𝑛, 𝑃˜𝑛 ≡ [0, 1] × ℙ𝑛 and define 𝜒𝑛 : 𝑃˜𝑛 → 𝑃𝑛 by 𝜒𝑛 (𝜆𝑛 , 𝑝0𝑛 , 𝑝1𝑛 ) =
(1 − 𝜆𝑛 )𝑝0𝑛 + 𝜆𝑛 𝑝1𝑛 . Then we have that ((𝑃˜𝑛 , ∂ 𝑃˜𝑛 ), (𝑃𝑛 , ∂𝑃𝑛 ), 𝜒𝑛 ) is a ball-bundle. Let 𝜒
be the product map 𝜒1 × 𝜒2 ; then ((𝑃˜ , ∂ 𝑃˜ ), (𝑃, ∂𝑃 ), 𝜒) is a ball-bundle too. Let 𝑌ˆ𝜀 =
[0, 𝜀] × 𝑃˜ . Then 𝜒 induces a map ℎ𝑌 : 𝑌ˆ1 → 𝑌1 by ℎ(𝜀, 𝜆, 𝑝0 , 𝑝1 ) = (𝜀, 𝜒(𝜆, 𝑝0 , 𝑝1 )). Now
((𝑌ˆ𝜀 , ∂ 𝑌ˆ𝜀 ), (𝑌𝜀 , ∂𝑌𝜀 ), ℎ𝑌 ) is a ball-bundle. Let 𝐸˜ be the set of all ((𝜀, 𝜆, 𝑝0 , 𝑝1 ), 𝑞) ∈ 𝑌ˆ1 × 𝑃
such that ℎ𝐸 (𝜀, 𝜆, 𝑝0 , 𝑝1 , 𝑞) ≡ (𝜀, 𝜒(𝜆, 𝑝0 , 𝑝1 ), 𝑞) ∈ 𝐸. Then also ((𝐸˜𝜀 , ∂ 𝐸˜𝜀 ), (𝐸𝜀 , ∂𝐸𝜀 ), ℎ𝐸 ) is a
ball-bundle. Moreover, letting 𝜃˜ be the projection from 𝐸˜ to ℙ, we have that ℎ𝑌 ∘ 𝜃˜ = 𝜃 ∘ ℎ𝐸 .
Therefore, by the Thom Isomorphism Theorem, 𝜃 is essential iff 𝜃˜ is; cf. [23, Appendix IV.3].
Let 𝐸ˆ be the closure of the set of (𝜀, 𝜆, 𝑝0 , 𝑝1 , 𝑞, 𝜋 1 (𝑞)) such that (𝜀, 𝜆, 𝑝0 , 𝑝1 , 𝑞) ∈ 𝐸˜ and
𝜆 ∕= 0. By the strong excision property, the natural projection 𝜙ˆ from 𝐸ˆ to 𝐸˜ induces
an isomorphism of their cohomology groups. Let 𝜃ˆ be the projection from 𝐸ˆ to 𝑌ˆ . Then
ˆ Therefore, 𝜃˜ is essential iff 𝜃ˆ is.
𝜃ˆ = 𝜃˜ ∘ 𝜙.
ˆ
Let 𝜂 : 𝐸ˆ → ℝ3+ be the projection map 𝜂(𝜀, 𝜆, (𝑝0 , 𝑝1 ), 𝑞, 𝜋 1 ) = (𝜀, 𝜆). Let 𝐷 = 𝜂(𝐸).
By the generic local triviality theorem, there exists a partition of 𝐷 into a finite number of
connected subsets 𝐷10 , . . . , 𝐷𝑙0 , and for each 𝐷𝑖0 a semi-algebraic fibre pair (𝐹𝑖 , ∂𝐹𝑖 ), a homeˆ such that 𝜂 ∘ ℎ𝑖 is the projection
omorphism ℎ𝑖 : 𝐷𝑖0 × (𝐹𝑖 , ∂𝐹𝑖 ) → (𝜂 −1 (𝐷𝑖0 ), 𝜂 −1 (𝐷𝑖0 ) ∩ ∂ 𝐸)
from 𝐷𝑖0 × 𝐹𝑖 to 𝐷𝑖0 . Since the sets 𝐷𝑖 are semi-algebraic, if necessary by decomposing them
AXIOMATIC EQUILIBRIUM SELECTION
41
into smaller sets, we can assume that the closure of each of these sets is homeomorphic to a
simplex. There now exists an 𝑖, say 1, such that the closure 𝐷𝑖 of 𝐷𝑖0 is homeomorphic to a
3-simplex and contains [0, 𝜀˜] × { (0, 0) } for some 𝜀˜ < 𝜀¯.
ˆ 1 ) be the closure of the inverse
Let (𝑌ˆ (𝐷1 ), ∂ 𝑌ˆ (𝐷1 )) ≡ (𝐷1 , ∂𝐷1 ) × (ℙ, ∂ℙ) and let 𝐸(𝐷
ˆ 1 ) : (𝐸(𝐷
ˆ 1 ), ∂ 𝐸(𝐷
ˆ 1 )) → (𝑌ˆ (𝐷1 ), ∂ 𝑌ˆ (𝐷1 )) be the
image of 𝐷1 × 𝐹1 under ℎ1 . Let 𝜃(𝐷
projection. We claim that (𝐸ˆ𝜀 (𝐷1 ), ∂ 𝐸ˆ𝜀 (𝐷1 )) is an orientable homology manifold (and hence
a pseudomanifold) for all small 𝜀, where 𝐸𝜀 is the inverse image under 𝜃𝐸𝜀 (𝐷1 ) of the points
(𝜀′ , 𝜆)𝑖𝑛𝐷1 with 𝑒′ ⩽ 𝜀. The set 𝐸ˆ was constructed from the set 𝐸, which is an orientable
homology manifold, by constructions involving ball bundles and homeomorphisms. Thus,
ˆ 1) ∖
𝐸ˆ is an orientable homology manifold. So our claim is proved if we show that 𝐸(𝐷
ˆ 1 ) is path-connected. There exist 0 < 𝜀ˆ′ < 𝜀˜ and integers 𝑟𝑛 > 1 for each 𝑛 such
∂ 𝐸(𝐷
that (𝜀, 𝜆) ∈ 𝐷1 if 0 < 𝜀 < 𝜀ˆ′ and 0 ≤ 𝜆𝑛 ≤ 𝜀𝑟𝑛 −1 . Now given 0 < 𝜀ˆ ⩽ 𝜀ˆ′ and given
two points 𝑥(0) and 𝑥(1) in 𝐸ˆ𝜀ˆ(𝐷1 ) ∖ ∂ 𝐸ˆ𝜀ˆ(𝐷1 ), connect them by a semi-algebraic curve
𝑥(𝑡′ ) = (𝜀(𝑡′ ), 𝜆(𝑡′ ), 𝑝0 (𝑡′ ), 𝑝1 (𝑡′ ), 𝑞(𝑡′ ), 𝜋 1 (𝑞(𝑡′ ))) in 𝐸ˆ𝜀¯ ∖ ∂ 𝐸ˆ𝜀¯ as 𝑡′ goes from 0 to 1. For each
𝑡′ , express 𝑞(𝑡′ ) as 𝜀(𝑡′ )(𝜆(𝑡′ )𝑝0 (𝑡′ ) + (1 − 𝜆(𝑡′ ))𝑝1 (𝑡′ ) + (1 − 𝜀(𝑡′ ))𝑟(𝑡′ ) where 𝑟(𝑡′ ) is a best
¯ ∗ that assigns to each 𝑡′ the set of 𝑞 ∗
reply to 𝑞(𝑡′ ). The correspondence from [0, 1] to 𝑄
such that 𝑟(𝑡′ ) is a best reply to 𝑞 ∗ is a nonempty, compact convex valued, and upper semicontinuous correspondence. Therefore, there exists a path ((𝑡′ (𝑡), 𝑞 ∗ (𝑡)) in the graph of this
correspondence with 𝑡′ (0) = 0 and 𝑡′ (1) = 1. We will now view the path 𝑥(𝑡′ ) as the path
𝑥(𝑡) ≡ 𝑥(𝑡′ (𝑡)).
Choose a positive 𝜀 such that 𝜀 + 𝜀𝑟𝑛 < 𝜀ˆ. For each 𝑛, modify 𝑥𝑛 (𝑡) to the vector
˜
𝑥˜𝑛 (𝑡) = (𝜀 + 𝜀𝑟𝑛 𝜀(𝑡), 𝜆(𝑡),
𝑝˜0 (𝑡), 𝑝1 (𝑡), 𝑞˜(𝑡), 𝜋 1 (𝑡)) ,
where
𝑟𝑛
˜ 𝑛 (𝑡) = 𝜀 𝜀(𝑡) 𝜆𝑛 (𝑡) ,
𝜆
𝜀 + 𝜀𝑟𝑛
𝑝˜0𝑛 (𝑡) =
𝜀𝑞 ∗ (𝑡) + 𝜀𝑟𝑛 𝜀(𝑡)(1 − 𝜆(𝑡))𝑝0 (𝑡)
,
𝜀 + 𝜀𝑟𝑛 𝜀(𝑡)(1 − 𝜆(𝑡))
𝑞˜𝑛 (𝑡) = (1 − 𝜀 − 𝜀𝑟𝑛 𝜀(𝑡))𝑟(𝑡) + 𝜀𝑞𝑛∗ (𝑡) + 𝜀𝑟𝑛 𝜀(𝑡)((1 − 𝜆(𝑡))𝑝0 (𝑡) + 𝜆(𝑡)𝑝1 (𝑡)) .
ˆ 1 ) ∖ ∂ 𝐸(𝐷
ˆ 1 ) for all 𝑡. Moreover, for 𝑡 = 0, 1, 𝑥(𝑡) and 𝑥˜(𝑡) can now
Then 𝑥˜(𝑡) belongs to 𝐸(𝐷
be connected by a path 𝑥ˆ(𝑡; 𝑠) defined as follows. For 𝑠 ∈ [0, 1], let 𝑘𝑛 (𝑠) = min(1, 2𝑠)𝑟𝑛 and
then:
ˆ 𝑠), 𝑝0 (𝑡; 𝑠), 𝑝1 (𝑡), 𝑞ˆ(𝑡; 𝑠), 𝜋 1 (𝑡)) ,
𝑥ˆ𝑛 (𝑡; 𝑠) = ((2𝑠 − 1)+ 𝜀 + 𝜀𝑘𝑛 (𝑠) 𝜀(𝑡), 𝜆(𝑡;
where
ˆ 𝑛 (𝑡) =
𝜆
𝜀𝑘𝑛 (𝑠) 𝜀(𝑡)𝜆𝑛 (𝑡)
,
(2𝑠 − 1)+ 𝜀 + 𝜀𝑘𝑛 (𝑠) 𝜀(𝑡)
42
SRIHARI GOVINDAN AND ROBERT WILSON
𝑝˜0𝑛 (𝑡; 𝑠) =
(2𝑠 − 1)+ 𝜀𝑞𝑛∗ (𝑡) + 𝜀𝑟𝑛 𝜀(𝑡)(1 − 𝜆𝑛 (𝑡))𝑝0𝑛 (𝑡)
,
(2𝑠 − 1)+ 𝜀 + 𝜀𝑘𝑛 (𝑠) 𝜀(𝑡)(1 − 𝜆𝑛 (𝑡))
𝑞˜𝑛 (𝑡; 𝑠) = (1−(2𝑠−1)+ 𝜀−𝜀𝑘𝑛 (𝑠) 𝜀(𝑡))𝑟𝑛 (𝑡)+(2𝑠−1)+ 𝜀𝑞𝑛∗ (𝑡)+𝜀𝑘𝑛 (𝑠) 𝜀(𝑡)((1−𝜆𝑛 (𝑡))𝑝0𝑛 (𝑡)+𝜆𝑛 (𝑡)𝑝1 (𝑡)) .
ˆ 1 ) ∖ ∂ 𝐸(𝐷
ˆ 1 ) is connected and hence an orientable homology manifold of dimension
Thus 𝐸(𝐷
ˆ 1 ) is
𝑑(ℙ) + 3. 𝑌ˆ (𝐷1 ) is a full-dimensional subset of 𝑌𝜀¯. Therefore 𝜃ˆ is essential iff 𝜃(𝐷
essential.
Since (𝐸ˆ𝜀ˆ(𝐷1 ), ∂ 𝐸ˆ𝜀ˆ(𝐷1 )) is an orientable homology manifold, (𝐹1 , ∂𝐹1 ) is now an orientable
homology manifold of dimension 𝑑(ℙ) and hence an orientable pseudomanifold. Moreover,
for each (𝜀, 𝜆) ∈ 𝐷10 , with 0 < 𝜀 ⩽ 𝜀ˆ, letting (𝐸ˆ𝜀,𝜆 , ∂ 𝐸ˆ𝜀,𝜆 ) ≡ ℎ−1
1 ({ 𝜀, 𝜆 } × (𝐹1 , ∂𝐹1 )) we have
that 𝜃ˆ is essential iff 𝜃ˆ𝜀,𝜆 , the projection map (𝐸ˆ𝜀,𝜆 , ∂ 𝐸ˆ𝜀,𝜆 ) → (ℙ, ∂ℙ), is essential.
ˆ
Let 𝐿 be the set of (𝜀, 𝜆) ∈ 𝐷1 such that 𝜆𝑛 = 𝜀𝑟 for some 𝑟 ⩾ 𝑟𝑛 for each 𝑛. Let 𝐸(𝐿)
be
ˆ
the closure of the inverse image of 𝐿 ∖ { (0, 0) } under ℎ1 . Let ∂ 𝐸(𝐿)
be the inverse image of
ˆ
𝐿 × ∂ℙ under the projection map 𝜃(𝐿) from 𝐸(𝐿)
to 𝐿 × ℙ. For each (𝜀, 𝜆) ∈ 𝐿, (𝐸𝜀,𝜆 , ∂𝐸𝜀,𝜆 )
is (𝜃(𝐿))−1 ({ (𝜀, 𝜆) } × (ℙ, ∂ℙ)). Our next objective is to show that (𝒬, ∂𝒬) equals the set of
(𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) such that (0, 0, (𝑝0 , 𝑝1 ), 𝑞 ∗ , 𝜋 1 ) belongs to 𝐸0,0 (𝐿). Given (0, 0, (𝑝0 , 𝑝1 ), 𝑞 ∗ , 𝜋 1 )
ˆ
ˆ
in 𝐸(𝐿)
there exists a sequence (𝜀(𝑘), 𝜆(𝑘), (𝑝0 , 𝑝1 )(𝑘), 𝑞(𝑘), 𝜋 1 (𝑘)) in 𝐸(𝐿)
∖ 𝐸0,0 converging
1 0
to it. For each 𝑛 and 𝑘, we can express 𝑞𝑛 (𝑘) as (1 − 𝜀(𝑘))((1 − 𝜇𝑛 )𝑞𝑛 (𝑘) + 𝜇1𝑛 𝑟𝑛1 (𝑘)) +
𝜀(𝑘)(𝜆(𝑘)𝑝0𝑛 (𝑘) + (1 − 𝜆(𝑘))𝑝1𝑛 (𝑘)), where 𝜇1𝑛 and 𝜆(𝑘) converge to zero, 𝑞𝑛0 (𝑘) belongs to 𝑃𝑛0
and converges to 𝑞𝑛∗ , and 𝑟𝑛1 (𝑘) belongs to 𝑃𝑛1 . Also 𝑞𝑛0 (𝑘) and 𝑟𝑛1 (𝑘) if 𝜇1𝑛 > 0 are best replies
to 𝑞(𝑘) for all 𝑘. By going to a subsequence, the faces to whose interior 𝑞𝑛0 (𝑘) and 𝑟𝑛0 (𝑘)
belong are constant for all 𝑘. The sequence generates for each 𝑛 an LPS Λ𝑛 = (¯
𝑞𝑛0 , . . . , 𝑞¯𝑛𝑙 )
where 𝑞¯𝑛0 = 𝑞𝑛∗ . As in the proof of Theorem 5.1, there exists a level 𝑙𝑛𝑖 for each 𝑖 = 0, 1 that is
expressible as a convex combination of 𝑝0𝑛 and another strategy. Since 𝜆(𝑘) converges to zero,
¯ ∗ for each 𝑙 < 𝑙0
𝑙𝑛0 < 𝑙𝑛1 . As in the proof of Theorem 5.1, we can prove that 𝑞¯𝑛𝑙 belongs to 𝑄
𝑛
𝑛
𝑙0
and if we express 𝑞¯𝑛𝑛 = 𝜈𝑛0 𝑝0𝑛 + (1 − 𝜈𝑛0 )𝑟𝑛0 , then the strategies 𝑞¯𝑛𝑙 for 𝑙 < 𝑙𝑛0 and 𝑟𝑛0 if 𝜈 0 < 1
𝑙1
are lexicographic best replies against Λ𝑚 . Likewise, if we express 𝑞¯𝑛𝑛 as 𝜈¯𝑛0 𝑝˜0𝑛 + 𝜈¯𝑛1 𝑝1𝑛 + 𝜈¯𝑛2 𝑟𝑛2 ,
then the strategy 𝑟𝑛2 if 𝜈¯𝑛2 > 0 is a lexicographic best reply against Λ𝑚 . As in the proof of
Theorem 5.1 in subsection 5.8, we can now write down an LPS (𝑞 ∗ , 𝑞¯0 (𝜀), 𝑞¯1 (𝜀)) to show that
(𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) belongs to 𝒬.
Given (𝑞 ∗ , (𝑝0 , 𝑝1 ), 𝜋 1 ) in 𝒬, it belongs to 𝒬(𝑇 ) for some 𝑇 . There exist 𝑞𝑛0 = (1 − 𝜆0𝑛 )𝑝0𝑛 +
𝜆0𝑛 𝑟𝑛0 ∈ 𝐵𝑛∗ (𝑇 ) and 𝑞𝑛1 = 𝜇0𝑛 𝑝˜0𝑛 + 𝜇1𝑛 𝑝1𝑛 + 𝜇2𝑛 𝑟𝑛2 ∈ 𝐶𝑛∗ (𝑇 ). As in the proof of Lemma C.9, we can
assume that the strategies in 𝑇𝑚 are best replies against the LPS (𝑞𝑛∗ , 𝑞𝑛0 , 𝑞𝑛1 ). For all small 𝜀,
choose 𝛼𝑛 (𝜀) such that 𝜀 = 𝜀𝑟+1 (1/𝜇1𝑛 )(𝜇0𝑛 +𝜇1𝑛 )+𝛼𝑛 (𝜀)(1−𝜆0𝑛 ), where 𝑟 satisfies the property
in the first line of the previous paragraph. Then for all small 𝜀, (𝜀, (𝜀𝑟 , 𝜀𝑟 ), (𝑝0 (𝜀), 𝑝1 ), 𝑞(𝜀))
AXIOMATIC EQUILIBRIUM SELECTION
43
ˆ
belongs to 𝐸(𝐿),
where for each 𝑛:
𝑝0𝑛 (𝜀) =
𝜀𝑟+1 (𝜇0𝑛 /𝜇1𝑛 )˜
𝑝0𝑛 + 𝛼𝑛 (𝜀)(1 − 𝜆0𝑛 )𝑝0𝑛
𝜀𝑟+1 (𝜇0𝑛 /𝜇1𝑛 ) + 𝛼𝑛 (𝜀)(1 − 𝜆0𝑛 )
and 𝑞𝑛 (𝜀) = (1 − 𝛼𝑛 (𝜀) − 𝜀𝑟+1 (1/𝜇1𝑛 ))𝑞𝑛∗ + 𝛼𝑛 (𝜀)𝑞 0 + 𝜀𝑟+1 (1/𝜇1𝑛 )𝑞𝑛1 .
For each (𝜀, 𝜆), (𝐸𝜀,𝜆 , ∂𝐸𝜀,𝜆 ) is a pseudomanifold. Indeed for (𝜀, 𝜆) ∕= (0, 0) this follows
from the fact that this pair is homeomorphic to (𝐹1 , ∂𝐹1 ), which is a pseudomanifold; for
(0, 0), this follows from the fact that (𝐸0,0 , ∂𝐸0,0 ) is homeomorphic to (𝒬, ∂𝒬). The inclusion
map (𝐸𝜀,𝜆 , ∂𝐸𝜀,𝜆 ) induces an isomorphism of the 𝑑(ℙ)-th cohomology groups. Thus 𝜃ˆ0,0 is
essential iff 𝜃ˆ𝜀,𝜆 is essential for some (and then all smaller) (𝜀, 𝜆) ∈ 𝐿. The projection map
𝜃ˆ0,0 is just the map Ψ. Hence 𝑄∗ is stable iff Ψ is essential.
□
In case 𝑆𝑛1 is empty, the construction is modified as follows. We can omit the sets 𝐶𝑛 (𝑇 )
and 𝐵𝑚 (𝑇 ) from the description of 𝒬(𝑇 ). In the last lemma above, the vector 𝜆 is now just
a number, one for player 𝑚. The simplex 𝐷 constructed there is 2-dimensional and contains
a curve 𝐿 of the form 𝜆 = 𝜀𝑟 . The rest of the proof is essentially the same.
This concludes the proof of the Theorem.
□
Appendix D. Construction of The Map 𝑔
If the projection map Ψ is inessential then there exists a continuous map 𝑔 : 𝒬 → ℙ that
has no point of coincidence with 𝑔. Therefore, there exists 𝛼 > 0 such that ∥Ψ(𝑥)−𝑔(𝑥)∥ > 𝛼
for all 𝑥 ∈ 𝒬.
Suppose now that Ψ is essential. Since 𝒬 is a pseudo-manifold of the same dimension as
ℙ, essentiality of Ψ in the sense we have defined it in Section 5 is equivalent to essentiality
of Ψ in cohomology [23, Theorem, Section 4][11, Lemmas A.3, A.4], i.e. Ψ∗ : 𝐻 𝑑 (ℙ, ∂ℙ) →
𝐻 𝑑 (𝒬, ∂𝒬) is nonzero. Moreover, letting 𝑑 be the dimension of ℙ and Ψ∂𝒬 the restriction of
Ψ to ∂𝒬, the degree of Ψ equals 𝛿 ∗ ∘ Ψ∗∂𝒬 (1), where 1 is the generator of 𝐻 𝑑−1 (∂𝒬) ≈ ℤ and
𝛿 ∗ is the coboundary operator.
Fix some 𝑝¯ in the interior of ℙ and define 𝜄 : ∂ℙ → ∂ℙ as follows: 𝜄(𝑝) is the unique point
in the boundary of the form 𝜆𝑝 + (1 − 𝜆)¯
𝑝 for 𝜆 < 0. 𝜄 is a homeomorphism without a fixed
point. Let 𝑔∂𝒬 : ∂𝒬 → ∂ℙ be the function 𝜄 ∘ Ψ∂𝒬 , where Ψ∂𝒬 is the restriction of Ψ to ∂𝒬.
Then 𝑔∂𝒬 has no point of coincidence with Ψ∂𝒬 , i.e. for each 𝑥 ∈ ∂𝒬, 𝑔∂𝒬 (𝑥) ∕= Ψ(𝑥). Also,
since 𝜄 is a homeomorphism, 𝛿 ∗ ∘ 𝑔∂𝒬 is nonzero.
∗
Construct a continuous map 𝑔∂𝑉 (𝑥∗ ) from ∂𝑉 (𝑥∗ ) to ∂ℙ such that 𝛿 ∗ ∘ 𝑔∂𝒬∪∂𝑉
(𝑥∗ ) (1) = 0
in 𝐻 𝑑 (𝑄 ∖ (𝑉 (𝑥∗ ) ∖ ∂𝑉 (𝑥∗ )), ∂𝒬 ∪ ∂𝑉 (𝑥∗ )). By the Hopf Extension Theorem [34, Corollary
8.1.18], the two maps 𝑔∂𝒬 and 𝑔∂𝑉 (𝑥∗ ) can be extended to a continuous map from 𝒬 ∖ (𝑉 (𝑥∗ ) ∖
∂𝑉 (𝑥∗ )) to ℙ; furthermore, by mapping points in 𝑉 (𝑥∗ ) to ℙ in a way that extends 𝑔∂𝑉 (𝑋 ∗ ) ,
we obtain a map 𝑔 : 𝒬 → ℙ such that all its points of coincidence with Ψ, of which there
44
SRIHARI GOVINDAN AND ROBERT WILSON
is at least one, are contained in 𝑉 (𝑥∗ ) ∖ ∂𝑉 (𝑥∗ ). There now exists 𝛼 > 0 such that for all
𝑥∈
/ 𝑉 (𝑥∗ ) ∖ ∂𝑉 (𝑥∗ ), ∥Ψ(𝑥) − 𝑔(𝑥)∥ > 𝛼.
Appendix E. The Genericity Assumption
The conclusions of this paper necessarily hold only when, fixing the game tree, the payoffs
lie in a generic set. Here we outline the nature of the genericity that is invoked. First, we
require that the game has finitely many equilibrium outcomes: in [5] we show that outside
a lower-dimensional set of payoffs every game has finitely many outcomes. Second, the
constructions in Appendix C rely on certain polyhedra being in general position. Each of
these polyhedra, of which there are finitely many, is a set of enabling strategies for a player
𝑛 against which, in a certain class of strategies for player 𝑚, a subclass is optimal. Since
these are defined by linear equations and inequalities in the payoffs of player 𝑚, the set of
games where the arguments fail is a lower-dimensional set. Third, Lemma C.11 requires
a characterization of stable sets that in [11] we show holds for all games outside a lowerdimensional set.
References
[1] Blume, L., A. Brandenburger, and E. Dekel (1991): Lexicographic Probabilities and Equilibrium Refinements, Econometrica, 59, 81-98.
[2] Blume, L. and W. Zame (1994): The Algebraic Geometry of Perfect and Sequential Equilibrium, Econometrica, 62, 783-794.
[3] Govindan, S., and T. Klumpp (2002): Perfect Equilibrium and Lexicographic Beliefs, International
Journal of Game Theory, 31, 229-243.
[4] Govindan, S., and J.-F. Mertens (2003): An Equivalent Definition of Stable Equilibria, International
Journal of Game Theory, 3, 339-357.
[5] Govindan, S., and R. Wilson (2001): Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes,
Econometrica, 69, 765-769.
[6] Govindan, S., and R. Wilson (2002): Structure Theorems for Game Trees, Proceedings of the National
Academy of Sciences USA, 99, 9077-9080. URL: www.pnas.org/cgi/reprint/99/13/9077.pdf
[7] Govindan, S., and R. Wilson (2002): Maximal Stable Sets of Two-Player Games, International Journal
of Game Theory, 30, 557-566.
[8] Govindan, S., and R. Wilson (2005): Essential Equilibria, Proceedings of the National Academy of
Sciences USA, 102, 15706-15711. URL: www.pnas.org/cgi/reprint/102/43/15706.pdf.
[9] Govindan, S., and R. Wilson (2006): Sufficient Conditions for Stable Equilibria, Theoretical Economics,
1, 167-206.
[10] Govindan, S., and R. Wilson (2008): “Nash Equilibrium, refinements of,” entry in S. Durlauf and L.
Blume (eds.), The New Palgrave Dictionary of Economics, 2nd Edition. New York: Palgrave Macmillan.
URL: gsbapps.stanford.edu/researchpapers/library/RP1897.pdf
[11] Govindan, S., and R. Wilson (2008): Metastable Equilibria, Mathematics of Operations Research, 33,
787-820.
[12] Govindan, S., and R. Wilson (2009): Axiomatic Theory of Equilibrium Selection in Games with Games
with Two Players, Perfect Information, and Generic Payoffs, Research Paper 2008, Stanford Business
School, Stanford CA. URL: gsbapps.stanford.edu/researchpapers/library/RP2008.pdf
[13] Govindan, S., and R. Wilson (2009): On Forward Induction, Econometrica, 77, 1-28.
AXIOMATIC EQUILIBRIUM SELECTION
45
[14] Hillas, J., T. Kao, and A. Schiff (2002): A Semi-Algebraic Proof of the Generic Equivalence of QuasiPerfect and Sequential Equilibria. University of Auckland, mimeo.
[15] Hillas, J., and E. Kohlberg (2002): Conceptual Foundations of Strategic Equilibrium, in R. Aumann
and S. Hart (eds.), Handbook of Game Theory, III, 1597-1663. New York: Elsevier.
[16] Kahneman, D., and A. Tversky (eds.) (2000): Choices, Values, and Frames. Cambridge, UK: Cambridge
University Press.
[17] Kohlberg, E., and J.-F. Mertens (1986): On the Strategic Stability of Equilibria, Econometrica, 54,
1003-1037.
[18] Koller, D., and N. Megiddo (1992): The Complexity of Two-Person Zero-Sum Games in Extensive
Form, Games and Economic Behavior, 4, 528-552.
[19] Kreps, D., and R. Wilson (1982): Sequential Equilibria, Econometrica, 50, 863-894.
[20] Kuhn, H. (1953): Extensive Games and the Problem of Information, in H. Kuhn and A. Tucker (eds.),
Contributions to the Theory of Games, II, 193-216. Princeton: Princeton University Press. Reprinted
in H. Kuhn (ed.), Classics in Game Theory, Princeton University Press, Princeton, New Jersey, 1997.
[21] Mailath, G., L. Samuelson, and J. Swinkels (1993): Extensive Form Reasoning in Normal Form Games,
Econometrica, 61, 273-302.
[22] Mertens, J.-F. (1989): Stable Equilibria–A Reformulation, Part I: Definition and Basic Properties,
Mathematics of Operations Research, 14, 575-625.
[23] Mertens, J.-F. (1991): Stable Equilibria–A Reformulation, Part II: Discussion of the Definition and
Further Results, Mathematics of Operations Research, 16, 694-753.
[24] Mertens, J.-F. (1992): The Small Worlds Axiom for Stable Equilibria, Games and Economic Behavior,
4, 553-564.
[25] Mertens, J.-F. (1993): Ordinality in Non Cooperative Games, International Journal of Game Theory,
32, 387-430.
[26] Nash, J. (1950): Equilibrium Points in N-Person Games, Proceedings of the National Academy of
Sciences USA, 36, 48-49.
[27] Nash, J. (1951): Non-Cooperative Games, Annals of Mathematics, 54, 286-295.
[28] Norde, H., J. Potters, H. Reijnierse, D. Vermeulen (1996): Equilibrium Selection and Consistency,
Games and Economic Behavior, 12, 219-225.
[29] Pimienta, C., and J.Shen (2010): On the Equivalence between (Quasi-) Perfect and Sequential Equilibria, University of New South Wales, mimeo.
[30] Reny, P. (1992): Backward Induction, Normal Form Perfection and Explicable Equilibria, Econometrica,
60, 627-649.
[31] Reny, P. (1992): Rationality in Extensive-Form Games, Journal of Economic Perspectives, 6, 103-118.
[32] Reny, P. (1993): Common Belief and the Theory of Games with Perfect Information, Journal of Economic Theory, 59, 257-274.
[33] Savage, L.J. (1954): Foundations of Statistics, New York: John Wiley & Sons.
[34] Spanier, E. (1966): Algebraic Topology, New York: McGraw-Hill. Reprinted New York: Springer-Verlag,
1989.
[35] van Damme, E. (1984): A Relation between Perfect Equilibria in Extensive Form Games and Proper
Equilibria in Normal Form Games, International Journal of Game Theory, 13, 1-13.
[36] van Damme, E. (2002): Strategic Equilibrium, in R. Aumann and S. Hart (eds.), Handbook of Game
Theory, III, 1523-1596. New York: Elsevier.
Department of Economics, University of Iowa, Iowa City IA 52242, USA.
E-mail address: [email protected]
Stanford Business School, Stanford, CA 94305-5015, USA.
E-mail address: [email protected]